A brief review of empirical equations for predicting heat transfer in the combustion chambers of steam boilers and petroleum heaters is followed by a study of eighty-five performance tests on nineteen furnaces differing widely in amount and arrangement of refractory cold surfaces. Operating conditions are available on furnaces with and without air preheat, with and without flue gas recirculation, fired with refinery cracked gas or oil fuel, and with a wide range of variation of excess air. The data are correlated by means of a theoretical equation and the deviations are no greater than the probable errors in the test data, and consistently less than those obtained by the empirical equation of Wilson, Lobo and Hottel. For simplicity of calculation the equation is presented in graphical form. An illustrative design problem has been included.

In view of the trend of the petroleum industry toward ever-increasing radiant heat transmission rates, as well as higher tube skin temperatures and higher percentages of heat-receiving surface per unit of refractory surface, this investigation has been initiated to study the effect of these variables and to find a means of allowing for their effect in the design of tubular oil heaters.

Eighty-five tests of nineteen different furnaces have been analyzed in this study. The test data include furnaces with and without air preheat and recirculation of flue gas. Excess air varied from 6% to over 170%, and average radiant rates from 3,000 to 54,000 Btu per hour per sq, ft. of circumferential tube area. The furnaces themselves were square, rectangular, or cylindrical in shape and varied widely in arrangement of surfaces; the ratio of effective refractory surface to equivalent cold plane surface varied from 0.45 to 6.55. Refinery cracked gas was the most common fuel, but a number of tests were made using oil fuel.

In this report a general and simple theoretical treatment is presented which satisfactorily correlates all the data. The deviations from the observed radiant section duties are well within the probable accuracy of the data. The average deviations of the predicted heat to the oil in the radiant section from the observed are 5.3% as compared to 6.85% when using the Wilson, Lobo, and Hottel empirical equation. The maximum deviation has been reduced from 33% to 16%. The data indicate that the larger deviations occurring when using the empirical equation are partly due to break-down of the equation below average radiant rates of 5,000 and above 30,000 Btu per hour per sq. ft. of circumferential area. It is likely that the empirical equation is seriously in error when applied to furnaces operating tube skin temperatures above 1000° F., as well as in furnaces having a low percentage of refractory surface and low values of PL, the product of partial pressure of the radiating constituents of the flue gas and the mean beam length of the radiating beam. The data available do not indicate any restriction which should be placed on the use of the theoretical equation herein presented.

*This paper is printed prior to presentation at the Thirty-Second Annual Meeting, Providence, Rhode Island, November 15, 16, and 17, 1939, in order to encourage discussion both verbal at the time of presenting and written.
Written discussion should be sent to the Secretary's Office promptly as it will be submitted to the author before publication. Written discussion received prior to November 10th will be read at time of presentation.
Discussion on this paper will be printed in Volume 36, No. 2, only.
!The M. W. Kellogg Company, New York City.

In view of the trend of the petroleum industry toward ever-increasing radiant heat transmission rates, as well as higher tube skin temperatures and higher percentages of heat-receiving surface per unit of refractory surface, this investigation has been initiated to study the effect of these variables and to find a means of allowing for them in the design of tubular oil heaters.

Although the exact mechanism of heat transmission in the radiant section of furnaces is complicated by factors about which little is known, certain generalizations and fundamental principles are fairly well established and can be used to advantage in solving radiant heat transfer problems. Some of these factors and their bearing on heat transfer problems are discussed below.

The major transfer of heat in the radiant section of a furnace is due to radiation from the hot gas cloud to the ultimate heat-receiving surface and by heat re-radiated from from the hot refractory surfaces to the cold surface. Some of the heat is also transferred at the instant of chemical union of the molecules in the flame. Of the radiation from the gas cloud, the major part is due to radiation from the carbon dioxide and water molecules present in it. Incandescent soot particles are a source of some radiation, but with fuels and burners commonly used in the petroleum industry, combustion usually results in a practically non-luminous flame. Oil fuels tend to give a more luminous flame than refinery gas at the usual percentages of excess air because of the cracking of the oil particles to soot during the combustion period. Data are not available on the exact degree of luminosity of oil flames, but it is probably a function of burner design, the amount of steam used in atomization and the percent excess air used.

In modern furnaces increasing amounts of heat are transmitted directly from the gas mass and lesser amounts are transmitted by the way of the refractory because the current trend is to fill the radiant section with cold tube surface in the interest of economy. Since the radiating constituents in the flue gas are the H₂O, CO₂ and SO₂ molecules present, the amount of heat radiated by them will be a function of their number and the temperature of the gas and cold surfaces. One measure of their number is their partial pressure. Another measure of their number is the mean length of the radiating beam in the gas mass. Hottel ¹ has shown that the product PL, atmospheres-feet, expresses these two facts and permits the data on the radiation from gases to be correlated. For any given fuel, P is a function of the excess air used and L is a function of the furnace alone. An equation, to be valid for a wide variety of sizes and shapes of furnaces, must take into account the effect of PL on furnace performance.

Heat transfer by convection to the tubes in the radiant section of petroleum heaters accounts for only a small amount of the heat transferred, especially in high radiant rate furnaces. This convection transfer is more important in low rate furnacesbecause heat transfer by convection is proportional to the temperature difference T_g - T_s, between flue gas and cold surface, whereas the radiant heat transfer is proportional to the difference T⁴_g - T⁴_s where the temperatures are expressed as degrees Rankine.

In view of the complexity of the problem, numerous investigators have correlated furnace performance by means of empirical equations. To illustrate the basic approach several of these empirical treatments are briefly summarized and their outstanding limitations described. A more complete review of this earlier lititure has been made in a previous publication.²

The following empirical equations have been classified into two major groups according to whether they are similar to the Hudson or the DeBaufre type equation.

Hudson³ correlated the data on several types of steam-boiler furnaces by the simple equation:

(1)

m = fraction of total heat input to the furnace (above the steam temperature) which is absorbed by the ultimate heat receiver.
G = air-fuel ratio, lbs. air / lb. fuel fired
C = pounds of fuel per hour per sq. ft. of water-cooled surface.

(2)

m = fraction of heat transferred above atmospheric temperature
G = air-fuel ratio, lbs. air / lb. fuel fired
C_o = pounds of equivalent good bituminous coal per hour per sq. ft. of water-cooled surface.

Wilson, Lobo and Hottel ² modified the Orrok equation and correlated the performance on ten of twelve furnaces. Their recommended equation is :

(3)

m = fraction of total heat input above 60°F. absorbed by the cold surface
a A_cp = effectiveness of tube surface as compared to a continuous cold plane, sq. ft.
Q = net heat liberated from combustion of the fuel, B.t.u. per hour
G = air-fuel ratio, lbs. air per lb. fuel fired.

Hottel ⁵ has proposed the following type of equation :

(4)

H = total net heat input from all sources, B.t.u. per hour
N = the hourly mean heat capacity of the flue gas between the temperature of the gas leaving the chamber and a base temperature of 60° F. B.t.u./hour/°F.
f = an overall exchange factor defined by the equation :

(5)

where
q = heat transferred by radiation, B.t.u./hour
T_g = temperature of the gas or hot surface, °F. + 460
T_s = temperature of cold surface, °F. + 460.

The overall exchange factor makes allowance for variation in effective flame emissivity, arrangement of refractory and non-black conditions in the furnace chamber. The constants in the above equation are very tentative so that it is only to be considered as illustrating a method. The f concept has been satisfactorily used in the equation presented in this paper and has been defined and discussed under the derivation of the theoretical equation.

DeBaufre ⁶ proposed an empirical equation which is similar to the basic Stefan-Boltzmann equation :

(6)

q = heat transferred, B.t.u./hour
A_o = total tube surface exposed to radiation, sq. ft.
T_g = temperature of the products of combustion leaving the furnace chamber, °F. + 460
T_s = temperature of cold surface, °F. + 460
E = effectiveness factor of the cold surface.

DeBaufre attempted to correlate E as a function of the rate of heat liberation per unit of furnace volume but the correlation was poor. For black body conditions E would have a maximum value of 0.173, the Stefan-Boltzmann constant.

Mekler ⁷ proposes the equation :

(7)

q = heat transferred by radiation, B.t.u./hour
S_e = equivalent "effective" heating surface, sq. ft.
C = an empirical coefficient depending on the temperature used for T_g
T_g = exit gas or theoretical flame temperature, °F. + 460
T_s = temperature of cold surface, °F. + 460.

In contrast to DeBaufre, however, he evaluates S_e as a function of the "fraction cold" of the furnace. An approximate graphical method is used for solving the DeBaufre type of equation. No cognizance is taken of the effect of PL on the heat transferred, and furnaces having the same geometric shape but widely different volumes are presumed to have the same fractional heat absorbtion. The effect of excess air on flame emissivity is likewise neglected.

If a series of furnaces operate on fuels whose heating value varies between comparatively narrow limits and whose ultimate heat-receiving surface temperatures are nearly constant, the performance of these furnaces may be adequately described by a simple empirical equation. However, as conditions deviate from those used to determine the constants of the equation its validity will be questionable. For example, a valid equation determined for tubes at a low temperature will certainly be invalid when the tube temperature is increased to a much higher value because as this temperature is increased it becomes more difficult to transfer a given amount of heat in a given furnace. In other words, to maintain a definite radiant rate in a furnace with a high tube temperature, more fuel must be fired.

Another disadvantage of the empirical equation is the difficulty of allowing for the effect of excess air unless a fuel of approximately constant heating value is used. If, for example, an air-fuel ratio is used to measure the effect of air addition, large values of the ratio are commonly associated with high percentages of excess air and low available heats. Conversely, small values of the ratio are associated with low percentages of excess air and high available heats. The danger involved in the indiscriminate use of an air-fuel ratio is best illustrated by an example. For theoretical combustion an average blast furnace gas would have an air-fuel ratio of approximately 0.73 lbs. air per lb. fuel gas and a natural gas might have a ratio of 15.9, yet the theoretical flame temperature of the blast furnace gas would be only 2800° F. as compared to 3580° F. for the natural gas. If the excess air in the case of the combustion of the blast furnace gas were increased until the air-fuel ratio became 15.9, the theoretical flame temperature would drop to a temperature much lower than 2800° F. If the the tube temperature was then increased, a point would be reached where no heat could be transferred to the tubes. However, with the same net heat liberation, natural gas fuel would transfer a finite amount of heat.

Realizing the limitations of the purely empirical approach it was decided to develop a theoretical radiant heat equation, simplified by assumptions, if necessary, and to test it by the application to^* data obtained on furnace performance.

*It should be remembered, however, that in many furnaces the usual measurement of the temperature of the gases leaving the radiant section does not give T_g directly, but a value usually less than T_g depending on the quantity of heat lost by the flue gases to the convection section by radiation at the point of measurement.

By a proper definition of terms the heat transferred in the radiant section could be predicted exactly by a Stefan-Boltzmann type equation.

(8)

q' = net heat transferred by radiation to the tubes, B.t.u./hour
T'_g = mean temperature of the hot gases in the furnace, °F. + 460
T'_s = mean tube skin temperature, °F. + 460. a A_cp = area of a plane which will absorb the same as the actual cold surface in the furnace, sq. ft.
f = an overall exchange factor correcting for flame emissivity, arrangement of the refractory, volume of the combustion chamber, etc. This factor will be discussed in detail later.

In the combustion chamber T_g, the mean temperature of the hot gases in the furnace and the temperature of the exit gases will undoubtedly differ, but run somewhat parallel. However, it was assumed that T'_g

All the net heat transferred to all the surfaces in the radiant section, i.e., the heat lost by the flame, is given by the following equation :

(9)

A'_r = area of refractory in furnace, sq. ft.
A_o = circumferential tube surface, sq. ft.
h_c = convection coefficient, B.t.u./hour/sq.ft./°F.

Since both the external losses from the furnace and the net heat transferred to the refractory by convection, given by the term h_cA'_r(T_g - T_r), are usually small, the two may be assumed equal without appreciably affecting the results. Equation (9) may then be rewritten to give instead the heat received by the oil :

(10)

The second term represents the heat transferred to the tubes by convection and it may be approximated as its magnitude is usually much smaller than the first term of equation (10).

By making the assumptions that :

	1. The convection coefficient lies normally between 2 and 3 B.t.u./hour/sq.ft./°F.;
	2. In most furnaces Ao equals (2a Acp) approximately;
	3. The overall exchange factor f has a value of about 0.57;

the terms h_c and A_o in equation (10) can be expressed in terms of a A_cp and f, thus :

(11)

Making this substitution in equation (10) :

(12)

Of the various ways that equation (12) representing the heat transfer relation can be combined with the equation representing a heat balance on the combustion chamber, the following graphical procedure is suggested :

Let

	H = the total net heat input to the furnace from all sources; i.e., combustion of the fuel, sensible heat in the air and fuel, sensible heat in recirculated flue gases, etc., B.t.u./hr.
	q = the total net heat absorbed in the radiant section by the ultimate heat-receiving surface, B.t.u./hr.
	N = the hourly heat capacity of the flue gas evaluated at the temperature of the gasses leaving the section, B.t.u./hr./°F. = (mols/hours) (mean MCp between tg and 60°F.) (see Figure No. 15 in Appendix)
	b = fraction of total net heat input lost from the external furnace walls
	tg = temperature of the gases leaving the section, °F.
	bH = total heat losses, B.t.u./hr.

The heat balance equation may be written : (Datum temperature = 60° F.)

(13)

By multiplying both sides of the equation by H(1 - b) we obtain :

(14)

Then

(15)

This may be transposed to :

(16)

The merit of such a relation is that the term in parentheses in the brackets is dependent on the fuel characteristics, excess air, air preheat, etc., and may be separately evaluated. After its value is determined, it may be used to establish the simple graphical relation among H((1 - b))/ aA_cp, q/aA_cp, t_g, in accordance with the following construction :

Since the term (t'_f - 60) is, from its method of construction the theoretical temperature the gases would attain (a) if combustion were adiabatic except for the loss of the fraction b of the enthalpy of the fuel and (b) if the products of combustion had a mean specific heat equal to their mean value from t_g down to the base temperature, the term t'_f may be thought of as a sort of flame temperature, referred to hereafter as the pseudo-flame temperature for short.

If now a plot is constructed with the coordinates q/aA_cp and t_g' a series of lines representing equation (12) may be placed on the plot; one for each value of surface temperature t_s (see Figures 1 and 2). On the same diagram a straight line through the value t'_f (completely determined by fuel characteristics, and presented for varying conditions of operation in Figures 3 and 4 in which the radiant section external losses have been taken as 2% of the total net heat input to the furnace) drawn through a point on the family of curves corresponding to the correct tube skin temperature, t_s, and for q/aA_cp or t_g (according to which of these is fixed) when extended to the left, will intersect the line t_g=60° at the value of H((1 - b))/aA_cp.

In design calculations the temperature, t_g, of the flue gases leaving the radiant section must be estimated before the pseudo-flame temperature can be obtained. This assumed or provisional t'_f must be revised if the assumed bridge wall temperature is found to be considerablely in error. The descriptive example included at the end of this section illustrates the method of calculation.

A_cp is the area of a continuous plane replacing the row of tubes and may be taken as the product of the exposed tube length, and center to center distance between tubes, and the number of tubes in the exposed radiant row. a is the ratio of reception by the actual surface to reception by a continuous plane. Then the term aA_cp is the tube area expressed as equivalent cold plane surface, i.e., the area of a plane which will absorb the same as the actual cold surface in the furnace.

Hottel⁸ gives a as a function of the ratio

as in Figure 5.

The following example illustrates the method of calculating a Acp.
Assume a radiant section of the following characteristics :
	Size of tubes	5" outside diameter
	Center to center distance of tubes	10"
	Exposed length of tube	30 ft.
	Total number of tubes	60
	Arrangement of tubes, 2 rows on equilateral triangular spacing
	Number of tubes per row	30
	a=0.984, i.e., total to 2 rows
	aAcp=0.984(750)=738 sq.ft. equivalent cold plane surface.

Fig. 5 Distribution of Heat to One or Two Rows of Tubes Mounted on

Refractory Wall and Irradiated from One Side.

Tubes on equalateral triangular centers; ordinate expressed on basis of heat transfered from a plane to a plane replacing tubes, or to infinite number of rows of tubes. These curves are a good approximation for tubes placed on rectangular or square centers

By definition, the emissivity of the flame is the ratio of the heat actually transmitted from the flame to the cold surface to the heat which would have been transmitted had the flame and the cold surface been perfect radiators. An illustrative example of this calculation is available in the literature.¹¹

Figure No. 6 gives P_f in terms of (P_CO2+P_H2O)L, t_g, and t_s for cracked gas fuel and a tube emissivity of 0.90. This plot is also a good approximation for fuel oil. The radiation cjharts of Hottel⁹ which were used in this calculation are included in the Appendix as Figures Nos. 13 and 14.

Values of P_CO2 + P_H2O and the air-fuel ratio for typical cracked gas and oil fuels have been plotted on Figure No. 7. The analysis of the fuels on which these calculations are based are indicated in the Appendix.

The mean length, L, of the radiant beam in the combustion chamber may be estimated from Table I.

The overall exchange factor, f, as defined by Hottel⁹, is then :

(18)

Where

(19)

A_f = Area of the flame bundle, sq. ft.
In commercial furnaces A_f may be considered equal to A_t, and equation (19) may be simplified to the form used in this study :

(20)

The exact evaluation¹⁰ of F_rc is rather tedious. In an effort to simplify the evaluation of this factor, more than twenty furnaces differing as widely as possible in design were studied, using the exact technique referred to above. It was found that for ratios of A_r/aA_cp from 0 to 1, the value of F_rc was adequately given by the ratio aA_cp/A_t. For ratios of A_r/aA_cp from 3 to 6.5, F_rc was very nearly equal to aA_cp/A_r. Figure No. 8 embodies these results and gives f directly as a function of the ratio A_r/aA_cp and the flame emissivity P_f.

Before discussing the results which prove the validity of the assumptions made in the development of the radiant equation, a descriptive example will be given to illustrate the use of the general method.

Find the total net heat input (i.e., the enthalpy of the incoming fuel and air above 60° F., water as vapor) to the following furnace :

The results of the investigation are summarized in Tables II and III. Table II gives the characteristics of the furnaces studied and the ratio of the actual heat to the oil to that calculated by the proposed theoretical equation, as well as by the empirical equation of Wilson, Lobo and Hottel. Sketches of the general types of furnaces studied are shown in Figures 16 through 22 in the Appendix. These should be considered as diagrammatic only.

Table II. - Characteristics of Furnaces and Tests
						Total	Total		Mean					Ratio:
		Tube		Circum	Effective	Furnace	Effective		Length
		Outside	Tube	ferential	Tube	Wall	Refractory	Ratio	Radiant	Air	Flue Gas		No.of	Actual/Calculated
Furnace		Diameter	Spacing	TubeArea	Area	Surface	Surface	AR/Acp	Beam	Preheat	Recirculation	Fuel	Tests	Heat to Oil
	(h)													(g)
Symbol	General	O.D.	C-C	Ac	aAcp	AT	AR		L					Empirical	Theoretical
Unit	Type	Inches	Inches	Sq.Ft.	Sq.Ft.	Sq.Ft.	Sq.Ft.		Feet					Equation	Equation
*1	A	5	10	2,389	1,340	3,080	1,740	1.30	14.3	Yes & No	Yes & No	Gas	16	1.03	1.02
*2	B	4	6.75	1,496	756	3,271	1,515	2.00	19.6	Yes	No	Oil	17	0.93	0.94
*3	C	5	10	2,945	2,255	3,855	1,600	0.71	17.0		No	Gas	10	0.95	0.99
4	B	5	10	2,394	1,343	4,277	2,934	2.18	17.8	Yes	No	Gas	7	0.94	0.97
*5	D	5	17.3(a)	4,443	2,303	3,362	1,059	0.46	24.0(c)	Yes	No	Gas	3	0.92	1.00
6(d)	E	5	9	3,060	1,608	3,174	1,566	0.97	12.8	No	No	Gas	2	1.02	0.99
7(d)#1	E	5.5	9.75	12,467(b)	3,610	6,560	2,950	0.82	22.2	No	No	Gas	1	1.18	1.11
7 #2	E	5	9.25	7,153	3,698	6,560	2,862	0.77	22.2	No	No	Gas	1	1.11	1.14
*8	F	4	8.75	284	216	1,628	1,412	6.55	11.2	No	No	Gas	7	1.04	0.98
9	B	5	8.75	4,775	1,844	2,976	1,132	0.61	14.7	Yes	No	Gas	4	1.04	1.03
10	G	4	8.75	1,347	465	1,875	1,410	3.04	11.2	No	No	Oil	3	0.94	0.99
11	G	4	8.75	1,466	498	1,608	1,110	2.23	8.28(e)	No	No	(f)	3	0.77	1.07
12	B	5	8.75	2,314	1,197	3,423	2,226	1.86	15.9	No	No	Oil	2	0.98	1.08
13	D	5	10.25	5,780	2,108	3,333	1,225	0.58	22.5(c)	Yes	No	Gas	1	1.02	0.99
14	D	5	10.25	5,780	2,108	3,049	941	0.45	22.5(c)	Yes	No	Gas	1	0.94	0.91
15	D	5	10.25	2,890	1,928	3,067	1,139	0.59	22.5(c)	Yes	No	Gas	1	0.99	0.93
16	D	5	10.25	5,780	2,108	3,049	941	0.45	22.5(c)	Yes	No	Gas	1	0.99	0.98
17	B	5	8.75	1,718	887	2,369	1,482	1.67	13.2	No	No	Gas	1	0.92	0.94
18	B	5	8.75	1,867	965	2,143	1,178	1.22	12.6	No	No	Gas	3	0.96	1.02
19	E	5.5	9.25	3,616	1,820	3,885	2,065	1.14	17.1	No	No	Oil	1	0.90	1.08
FOOTNOTES :
	(a)	Two rows with center lines 2.62 inches apart.
	(b)	Double row of tubes.
	(c)	Circular furnace, L = diameter.
	(d)	Double radiant type furnace.
	(e)	L = 1.8 (minimum distance).
	(f)	Products of combustion from furnace No. 10.
	(g)	Wilson, Lobo, and Hottel Empirical Equation.
	(h)	For sketches of general type of furnace see Figs. 16 through 22 in the Appendix.
	*	Furnaces used by Wilson, Lobo and Hottel (2).

Table III. - Test Data and Calculations
					(b)	RadiantSection		(c)							Rate of Heat		Ratio:Actual/Calculated
			Average	(a)	Bridge	Temperatures		Average			(d)			(e)	Input	Absorption	Heat to Oil
			Radiant	Excess	Wall			Tube		Overall	TotalNet	Hourly	Pseudo-	Heat			(g)
Symbol	Furnace		RateTo	Air	Temper-	Oil	Oil	Skin	Flame	Exchange	Heat	Heat	Flame	into	0.98H	q	Empirical	Theoretical
Unit	Fuel	Test	FirstRow		ture	In	Out	Temp.	Emissivity	Factor	Input	Capacity	Temp.	Oil	aAcp f	aAcp f	Equation	Equation
			B.t.u./hr./		t'g			ts			H	N	t'f	q
			Sq.Ft.	%	°F.	°F.	°F.	°F.	Pf	f	106B.t.u./hr.	Btu/hr/°F	°F.	106B.t.u./hr.	106B.t.u./hr.
1	Gas	1	4,760	17.3	1,140	728	850	820	0.525	0.675	53.04	34,643	1,562	11.4(f)	57.40	12.60	1.04	1.01
		2	6,440	113.8	1,220	700	865	817	0.415	0.585	42.21	21,528	1,980	15.4	52.80	19.60	1.04	1.14
		3	7,190	80.8	1,250	680	868	805	0.456	0.621	37.68	16,692	2,275	17.2	44.50	20.60	1.24	1.12
		4	7,650	72.3	1,250	675	875	809	0.460	0.625	37.55	15,246	2,475	18.3	43.80	21.80	1.12	1.11
		5	9,110	51.0	1,230	645	875	788	0.480	0.645	35.25	10,196	3,455	21.8	40.40	25.35	1.07	1.06
		7	8,900	48.9	1,180	648	872	794	0.494	0.650	36.18	9,672	3,720	20.8	40.60	23.35	0.99	0.94
		8	7,990	32.4	1,180	660	870	794	0.514	0.665	36.57	11,942	3,060	19.1	40.35	21.43	1.01	0.97
		9	7,480	61.1	1,200	660	858	790	0.480	0.640	35.86	11,548	3,100	17.9	40.70	20.85	1.05	0.93
		10	6,440	122.0	1,215	680	850	795	0.419	0.590	38.05	17,211	2,225	15.4	47.10	19.43	0.98	1.04
		11	6,260	141.3	1,225	685	850	797	0.412	0.585	39.22	16,438	2,395	15.0	49.10	19.10	1.04	0.92
		12	5,440	178.0	1,225	700	843	803	0.395	0.565	44.32	22,715	1,970	13.0	57.00	17.07	0.94	0.93
		15	9,560	33.3	1,360	650	858	780	0.498	0.655	44.46	14,954	2,960	22.9	49.60	26.50	1.04	1.06
		16	11,030	30.5	1,440	650	885	793	0.485	0.641	51.22	16,889	3,040	26.4	58.40	30.75	0.99	1.05
		17	10,700	26.0	1,360	655	885	800	0.505	0.660	47.49	16,499	2,880	25.6	52.70	29.00	1.01	1.12
		18	6,470	24.4	1,250	728	865	822	0.520	0.670	65.32	36,228	1,823	15.5	71.15	17.24	0.91	0.95
		19	10,200	42.1	1,270	655	875	795	0.485	0.642	45.72	12,843	3,550	24.4	52.00	28.35	0.95	0.96
															Avg	....	1.03	1.02
2	Oil	1	7,160	114	1,270		681(b)	717	0.49	0.720	24.58	10,300	2,395	10.73	44.10	19.72	0.99	1.00
		2	4,200	184	1,260		688	715	0.501	0.780	20.94	11,300	1,873	6.30	37.10	11.70	0.77	0.87
		3	4,840	154	1,225		689	717	0.494	0.725	19.55	9,560	2,065	7.24	34.90	13.18	0.87	0.91
		4	7,720	38	1,415		692	726	0.585	0.772	23.55	7,640	3,080	11.58	39.40	19.78	0.91	0.90
		5	7,520	30	1,410		694	725	0.643	0.800	22.16	7,060	3,130	11.24	35.80	18.59	0.92	0.91
		6	7,890	51	1,450		702	735	0.570	0.770	25.46	8,690	2,935	11.80	42.90	20.20	0.91	0.89
		7	7,600	74	1,420		703	733	0.556	0.763	26.71	10,180	2,535	11.39	45.20	19.70	0.89	0.94
		8	7,590	83	1,440		704	735	0.541	0.750	27.78	10,540	2,640	11.36	47.70	20.00	0.89	0.89
		9	4,800	175	1,175		699	730	0.512	0.735	19.28	10,000	1,950	7.18	33.80	12.89	0.94	0.98
		10	5,100	167	1,205		702	736	0.502	0.730	20.23	10,200	2,002	7.64	35.95	13.82	0.94	0.97
		11	4,920	69	1,200		680	716	0.585	0.772	14.58	5,650	2,590	7.37	24.40	12.61	0.94	0.95
		12	5,180	48	1,205		677	717	0.610	0.785	14.30	5,080	2,820	7.75	23.60	13.02	0.95	0.96
		13	5,050	100	1,185		692	728	0.557	0.763	16.13	6,860	2,363	7.55	27.40	13.08	0.95	0.96
		14	5,320	120	1,225		698	735	0.517	0.740	17.84	7,820	2,300	7.96	31.20	14.20	0.96	0.97
		15	6,500	106	1,270		694	726	0.525	0.743	23.40	10,120	2,323	9.73	40.70	17.30	0.94	0.96
		16	6,780	42	1,230		679	716	0.584	0.770	18.54	6,340	2,920	10.14	31.15	17.40	0.99	0.99
		17	6,740	68	1,300		684	720	0.561	0.768	20.64	7,630	2,720	10.09	35.15	17.48	0.97	0.96
															Avg	....	0.93	0.94
3	Gas	1	8,360	55	1,390	749	927	884	0.475	0.557	46.93	15,300	3,060	24.66	36.60	19.60	0.98	0.99
		2	8,000	105	1,430	765	923	889	0.425	0.517	56.41	22,700	2,493	23.61	47.40	20.25	0.95	0.99
		3	9,010	35	1,490	758	947	900	0.492	0.570	50.32	14,900	3,380	26.61	38.40	20.75	0.94	0.94
		4	9,160	33	1,490	762	944	900	0.495	0.573	51.86	15,820	3,270	27.03	39.40	20.95	0.93	0.95
		5	8,860	66	1,430	764	944	898	0.450	0.538	56.90	21,100	2,700	26.17	46.00	21.58	0.94	1.00
		6	11,850	89	1,455	752	948	897	0.490	0.570	64.96	20,300	3,200	34.93	49.50	27.18	1.03	1.04
		7	10,520	50	1,440	749	948	896	0.475	0.557	62.22	21,050	2,960	31.04	48.60	24.73	0.98	1.02
		8	9,650	55	1,430	746	948	894	0.474	0.556	60.56	21,200	2,860	29.36	47.40	23.40	0.96	1.01
		11	6,760	119	1,400	742	878	853	0.410	0.505	49.03	20,620	2,395	19.92	42.20	17.50	0.89	0.96
		12	7,990	92	1,450	753	878	857	0.430	0.520	52.50	19,920	2,645	23.56	43.90	20.08	0.93	0.98
															Avg	....	0.95	0.99
4	Gas	1	7,840	44	1,454	711	937	842	0.492	0.740	38.33	12,254	3,120	18.78	36.95	18.90	0.95	0.92
		2	7,990	62	1,432	718	914	833	0.466	0.720	41.84	14,698	2,840	19.13	42.40	19.80	0.97	0.92
		3	7,160	71	1,401	714	909	826	0.466	0.720	39.09	14,408	2,295	17.17	40.40	17.70	0.94	1.04
		4	6,360	72	1,389	712	906	823	0.466	0.720	34.65	12,841	2,290	15.25	35.10	15.79	0.92	1.02
		5	9,900	75	1,454	749	937	871	0.466	0.720	54.94	20,772	2,335	23.68	55.50	24.50	1.02	1.14
		6	7,340	78	1,463	740	933	859	0.455	0.710	44.59	17,127	2,550	17.58	46.90	18.42	0.89	0.88
		7	7,550	81	1,472	716	934	847	0.455	0.710	46.11	18,009	2,560	18.09	47.50	18.96	0.90	0.89
															Avg	....	0.94	0.97
5	Gas	1	4,510	101	1,466	803	1,001	949	0.485	0.530	50.57	20,627	2,465	21.11	40.75	17.27	0.89	0.97
		2	4,800	103	1,473	825	1,025	969	0.485	0.530	54.41	22,310	2,450	22.40	44.00	18.30	0.91	0.98
		3	5,610	88	1,477	836	1,049	988	0.495	0.535	57.96	22,040	2,625	26.27	46.00	21.25	0.97	1.04
															Avg	....	0.92	1.00
6	Gas	1	14,910	5.1	1,781	666	1,004	986(j)	0.470	0.595	88.56	22,740	3,840	45.66	90.60	47.80	1.04	1.00
		2	14,840	6.4	1,789	664	1,006	990	0.470	0.595	90.04	23,344	3,840	45.46	92.10	47.40	1.03	0.98
															Avg	....	1.02	0.99
7	Gas	1	12,053	22	1,507	672	920	845	0.586	0.660	184.7	51,910	3,550	104.6	75.00	43.80	1.18	1.11
		2	9,300	41	1,433	753	948	903	0.570	0.650	138.6	43,875	3,160	74.1	52.40	30.85	1.11	1.14
															Avg	....	1.15	1.13
8	Gas	1	16,970(i)	128		598	818	880	0.323	0.762	21.45	16,958	2,090	4.82(f)	127.90	29.30	0.86	0.96
		2	23,800	64		510	788	872	0.365	0.788	19.85	11,491	2,750	6.77	114.20	39.80	1.01	0.95
		3	28,800	50		494	778	884	0.375	0.795	23.05	11,582	2,970	8.16	131.80	47.50	1.04	0.94
		4	36,700	34		480	789	929	0.370	0.790	30.28	16,994	3,290	10.43	174.00	61.10	1.04	0.91
		5	39,000	52		497	786	941	0.360	0.780	34.40	20,122	2,950	11.10	200.00	65.80	1.11	1.00
		6	42,500	53		512	786	958	0.360	0.780	38.44	18,442	2,870	12.09	224.00	71.60	1.10	1.07
		7	50,900	33		473	775	980	0.368	0.788	44.72	18,163	3,220	14.45	258.00	85.00	1.09	1.00
															Avg	....	1.04	0.98
9	Gas	1	12,470	26	1,725	749	978	990(j)	0.480	0.545	81.56	21,200	3,823	44.60	79.50	44.50	1.05	1.03
		2	12,450	32	1,652	745	973	995	0.471	0.540	83.01	23,000	3,600	44.63	81.60	44.90	1.08	1.07
		3	11,900	16	1,786	741	977	1,000	0.480	0.545	79.07	20,150	3,910	42.71	77.00	42.50	1.01	1.00
		4	11,530	46	1,711	759	975	1,007	0.445	0.516	85.90	25,850	3,320	41.48	88.30	43.50	1.03	1.04
															Avg	....	1.04	1.03
10	Oil	1	21,200	26	1,705	670	914	898	0.430	0.750	30.99	9,252	3,340	14.55	87.00	41.65	0.95	0.99
		2	20,400	21	1,695	677	912	898	0.437	0.753	28.47	8,187	3,475	13.98	79.90	39.83	0.95	0.98
		3	21,600	38	1,740	691	910	900	0.420	0.742	34.57	11,168	3,100	14.82	98.00	42.90	0.93	1.00
															Avg	....	0.94	0.99
11	(k)	1	3,140	111	1,037	573	670	682	0.390	0.665	15.82	11,701	1,388	8.67	46.90	11.10	0.67	1.08
		2	4,320	74	1,038	570	677	686	0.425	0.691	13.92	8,312	1,702	5.07	39.65	14.72	0.87	1.16
		3	3,840	91	1,103	593	691	703	0.395	0.667	19.06	12,276	1,585	5.35	56.25	13.51	0.77	0.98
															Avg	....	0.77	1.07
12	Oil	1	10,780	45	1,457	474	750	716	0.490	0.706	48.48	16,209	2,990	24.88	56.10	29.50	0.95	1.04
		2	8,770	63	1,399	470	729	689	0.475	0.695	41.66	15,531	2,690	20.03	49.20	24.10	0.91	1.02
															Avg	....	0.93	1.03
13	Gas	1	9,300	17	1,560	712	915	897	0.590	0.631	56.36	13,786	4,060	34.53	41.45	25.90	1.02	0.99
14	Gas	1	7,060	28	1,700	738	903	897	0.560	0.595	46.10	11,626	3,940	26.09	36.00	20.80	0.94	0.91
15	Gas	1	9,150	15	1,750	734	901	894	0.580	0.630	52.46	12,080	4,310	31.01	42.35	25.50	0.99	0.93
16	Gas	1	9,300	26	1,623	704	898	883	0.565	0.595	58.46	14,630	3,980	34.45	45.60	27.45	0.99	0.98
															Avg	....	0.98	0.96
17	Gas	1	9,810	59	1,505	712	892	872	0.415	0.630	42.19	15,000	2,815	16.89	74.00	30.20	0.92	0.94
18	Gas	1	9,330	57	1,480	737	950	925	0.417	0.580	39.99	14,062	2,850	17.41	70.10	31.20	0.97	1.00
		2	9,350	59	1,482	735	960	930	0.417	0.580	40.60	14,402	2,820	17.46	70.50	31.20	0.96	1.01
		3	8,310	86	1,415	685	905	870	0.395	0.560	39.79	16,222	2,460	15.52	72.10	28.75	0.95	1.05
															Avg	....	0.96	1.02
19	Oil	1	9,120	95	1,382	825	850	879	0.45	0.596	87.50	37,350	2,360	32.97	79.00	30.38	0.90	1.08
														Grand	Avg	....	0.969	0.992

(a)	At bridge wall.	Average Deviation	................	6.85 %	5.30 %
(b)	With high velocity thermocouples.	Maximum Deviation	................	33 %	16 %
(c)	Estimated from average oil temperature, inside coefficient with allowance for coke deposit.	% of Tests Between	0 and 2 % Deviation....	17.7 %	31.8 %
(d)	Net heat input from combustion of the fuel, air preheat, and recirculation if any.		0 and 4 % Deviation....	36.5 %	53.0 %
(e)	Based on flue gas duty, except for Furnaces Nos. 1 and 8.
(f)	Estimated from oil side. Applies only to furnaces Nos. 1 and 8.
(g)	Wilson, Lobo, and Hottel Empirical Radiant Equation.
(h)	Temperatures for Furnace No. 2 are average oil temperatures.
(i)	Based on exposed tube surface.
(j)	Actual tube skin thermocouple measurements.
(k)	Fuel = Products of combustion from Furnace No. 10.

The actual heat to the oil is obtained by subtracting the enthalpy of the flue gas leaving the radiant section from the enthalpy of the entering fuel and air, and then subtracting the estimated external losses, plus the direct radiant heat from the combustion chamber absorbed by the first rows of the convection section. The latter item is obtained by a convection section heat balance , gas side versus oil side, in which the heat lost by the gas is considered between the actually measured true gas temperature of the gases entering and leaving the section. A summary of the test data required and a short outline of the method of calculation are included in the Appendix.

Table III gives a summary of the test data and calculations for all eighty-five tests which have been used as an independent check on the validity of the derived theoretical equation. The ratio of the actual heat absorbed by the oil in the radiant section to that predicted by both the empirical and theoretical equations is given for each of the individual tests. The deviations of any test from the average deviations for a given furnace can thus be readily seen.

DISCUSSION OF RESULTS

In order to arrive at a definite conclusion regarding the respective merits of the Wilson, Lobo and Hottel empirical equation and the theoretical equation, the heat absorbed by the oil in the radiant section as predicted by these two equations has been compared to the actually observed heat absorption.

The radiant section will be defined as that section in which the heat is liberated and in which the heat transfer is primarily by radiation from the hot gas mass and the hot refractory surfaces. The cold or ultimate heat-receiving surface is considered to consist of those tubes which can "see" to a greater or less extent the main gas mass. This emphasis has been made because in the case of certain furnaces the first rows of tubes in the convection section can see the main gas mass. The plane area * of these tubes, A_cp, must be considered as a part of the radiant section equivalent cold plane surface, a A_cp. In furnaces where the convection section is placed behind a bridge wall, out of sight of the main gas mass, the projected area of the convection section is not considered to be part of the radiant section cold surface.

---

*In a bank of three or more rows of tubes a = 1.0, and A_cp = aA_cp.

It may be seen from Table III that in practically every case the equation presented in this paper correlates the test data better than the empirical equation even though the furnace tests used to determine the constants of the empirical equation are included in the present data. The maximum deviation has been reduced from 335 to 16%. These facts are perhaps best illustrated by Figures Nos. 9 and 10, which show the spread of the data using both methods of correlation. Inasmuch as the data represent plant tests, it is suspected that the larger deviations may be due to inaccuracies in the data and not to any fundamental fault in the radiant equation. It is significant that using the theoretical equation the greatest deviations are not confined to any one furnace but seem to be well distributed. Again, furnace No. 4, the worst case, shows deviations of -11% and -12% and +14%, indicating that the data on this one furnace are probably less reliable than the average. Actually, the temperature of the gases leaving the radiant section of this furnace was determined at only two points instead of by a complete traverse. Since in many cases it is difficult to judge exactly the accuracy of the plant data, no attempt has been made to segregate and give more weight to the better data. The number of tests available for this study is hardly large enough for a rigid statistical analysis, but for some generalizations are justified. For instance, considering the data in their entirety, Figures Nos. 10 and 12(A) indicate that the deviations between the observed and the calculated heat absorption occur in a random manner and are not peculiar to any one fiurnace, even though the ratio of effective refractory to effective cold surface, (A_r/aA_cp), varied fourteen-fold for the furnaces investigated.

Figure 11 (B) (C) (D) and Figure 12 (B) (C) (D) are given in order to permit visualizing the effect of some of the variables on the results obtained by the two correlations.

The effect of excess air is shown in Figures 11 (B) and 12 (B). It should be remembered that although the percentage of excess air varied less than 10% to more than 170%, other factors may likewise have been varying at the same time. In general, the empirical equation appears to predict correctly the heat absorption by the oil in the radiant section in the excess air range from 10% to 80%, the more usual commercial range. Above 80% it predicts heat absorptions which are 10% to 13% too high. Over this range, 10% to 170%, the theoretical equation shows no significant trend. As previously pointed out, the use of an air-fuel ratio is open to criticism and the trend may be due to this factor.

Figures 11 (C) and 12 (C) classify the data according to average radiant rate based on circumferential tube surface, B.t.u. per hour per sq. ft. In those furnaces with a double row of radiant tubes, the rate has been taken as the average first row rate. Here, again, the theoretical equation shows no significant trend over the range 3,000 to 51,000 B.t.u. per hour per sq. ft. The empirical equation shows signs of breaking down at rates below 4,000 and predicts radiant section heat absorption 10% in excess of the true figure at that point. Only two tests are available at rates between 3,000 and 4,000 but they confirm the trend and indicate that the heat absorbed by the oil, as calculated by the empirical equation, may be almost 40% high in this range.

Figures 11 (D) and 12 (D) give another indication of the range of the furnace data, but have significance only in that they give an indication of the theoretical flame temperature. These figures show that over the wide range of t'_f, the pseudo-theoretical flame temperature, the deviations have no definite trend. It is obvious, therefore, that petroleum heaters can be designed by means of the new equation for fuels of widely different heating values as a low pseudo-flame temperature may mean either a fuel of low grade, or a high heating value fuel degraded with excess air or recirculated inert gas. Since most of these tests have been made with very similar fuels, the trend shown in Figure 11 (D) is probably due to the composite effect of rate and excess air discussed above.

CONCLUSIONS AND RECOMMENDATIONS

The theoretical radiant equation as developed in this paper is recommended for the solution of heat transfer problems in the radiant section of tubular heaters using fuel oil or gaseous fuel.

The equation is believed to be valid for any condition of air preheat, inert gas recirculation, percentage of excess air, or radiant rate. The effect of furnace volume on the amount of heat liberation necessary to maintain any given radiant rate is adequately handled by the use of the flame emissivity as outlined previously. The results indicate that the f plot represents an accurate and simple method of simultaneously allowing for the effect of flame emissivity and the amount of refractory surface present in the radiant section.

The Wilson, Lobo, and Hottel empirical equation is not recommended for use on furnaces differing widely from those used to determine the constants of the equation. However, the equation may be used with safety on box-type heaters (see Figures 16, 17, 18, 20 and 21) when the greatest accuracy is not required and subject to the following qualifications :

1.	Fuel oil or cracked refinery gas as fuel;
2.	Radiant rates between 5,000 and 30,000 B.t.u./hr./sq.ft. of circumferential tube area ;
3.	Per cent excess air between 5% and 80%;
4.	Tube skin temperatures not closer than 400� F, to the temperature of the flue gas leaving the radiant section ;
5.	Length of the radiant beam, L, greater than 15 feet.

NOMENCLATURE

Ao	=	total outside tube area exposed to radiation, sq.ft.
AR	=	effective refractory area, sq.ft.
A'R	=	actual refractory area, sq.ft.
AT	=	total wall area in combustion section, sq.ft.
Acp	=	area of plane replacing tubes, sq.ft.
C	=	actual firing rate of fuel/sq.ft. of exposed tube area, lbs./hr./sq.ft.
Co	=	equivalent firing rate of good bituminous coal/sq.ft. of projected tube area, lbs./hr./sq.ft.
CA	=	firing rate based on projected tube area, lbs./hr./sq.ft.
E	=	effectiveness factor of the cold surface
Frc	=	fraction of all the radiation emitted from all the refractory in all directions, which, if not absorbed by the gas, would hit cold surface, aAcp
FS	=	angle-emissivity factor
G	=	air-fuel ratio, lbs. air /lb. of fuel
H	=	total net heat input to combustion chamber, B.t.u./hr.
hc	=	convection coefficient, B.t.u./hr./sq.ft./�F.
L	=	mean length of radiant beam, feet
N	=	hourly heat capacity, B.t.u./hr./�F.
Pe	=	emissivity of tube surface
Pf	=	emissivity of flame
PCO2	=	partial pressure of CO2, atmospheres
PH2O	=	partial pressure of water vapor, atmospheres
MCPavg.	=	mean molal heat capacity between 60� and temperature, tg
Q	=	net heat liberated from combustion of the fuel, B.t.u./hour
q	=	heat transferred to oil, B.t.u./hour
q'	=	heat transferred by radiation, B.t.u./hour
q"	=	net heat transferred to all surfaces in the radiant section, B.t.u./hour
Se	=	equivalent "effective" heating surface, sq.ft.
Tg	=	temperature of products of combustion leaving combustion chamber, �F. + 460�
Ts	=	tube skin temperature, �F. + 460�
t	=	temperature, �F.
tg	=	temperature of flue gas leaving combustion chamber, �F.
t'f	=	Pseudo-flame temperature, �F.
a	=	factor by which Acp must be reduced to obtain effective cold surface, aAcp (effective tube area)
b	=	fraction of total net heat input lost from the external furnace walls
f	=	overall exchange factor
m	=	fraction of heat available above 60� F. absorbed by cold surfaces in the combustion chamber

LITERATURE CITED
1.	Hottel, Trans. A.I.Ch.E. 19, 173 (1927); Ind. Eng. Chem. 19, 888 (1927)
2.	Wilson, Lobo, Hottel, Ind. Eng. Chem. 24, 486 (1932)
3.	Hudson, Engineer 70, 523 (1890)
4.	Orrok, Trans. A.S.M.E. 1148 (1925)
5.	Hottel, Unpublished notes on Radiant Heat Transmission, Mass. Inst. of Tech. (1938)
6.	DeBaufre, Trans. A.S.M.E. 53 (14), 253 (1931)
7.	Mekler, Nat. Pet. News 30 (30), R355 (1938)
8.	Hottel, Trans. A.S.M.E., Fuels Steam Power 53 (14), 265 (1931)
9.	Hottel, Unpublished notes on Radiant Heat Transmission, Mass. Inst. of Tech. (1938)
10.	Hottel, in Chem. Eng. Handbook 1, 888 (1934)
11.	Hottel, Ibid. 1, 910 (1934)
12.	Hottel, Ibid. 1, 892 (1934)

ACKNOWLEDGMENT

The authors gratefully acknowledge the helpful suggestions offered during the course of the investigation by Mr. J. H. Rickerman.

APPENDIX STANDARD FUEL COMPOSITION STANDARD CRACKED GAS FUEL

Component		Mol PerCent
CH4	...................................	37.0
C2H4	...................................	4.0
C2H6	...................................	22.0
C3H6	...................................	8.0
C3H8	...................................	23.0
C4H10	...................................	6.0
		____
		100.0

Lower heating value per lb. at 60� F. 20,557 B.t.u. (Higher heating value 22,400 B.t.u.)
Specific gravity relative to air, 1.06
Specific volume at 60� F. and 14.7 lb./sq.in. 0.0807 lb./ft.³
Combustion calculations are based on air at 60� F. and 50% relative humidity.

OIL FUEL

Component		Wt. PerCent
C	...................................	85.0
H2	...................................	12.0
S	...................................	1.5
O2	...................................	0.7
N2	...................................	0.8
		____
		100.0

Lower heating value per lb. at 60� F. 17,130 B.t.u.
Steam for atomization, 0.3 lb./lb. fuel oil
Air at 60� F. and 50% relative humidity.

TEST DATA REQUIRED FOR D ETERMINATION OF HEAT ABSORBED BY OIL IN RADIANT SECTION

A.	Quantities:
	1.	Fuel.
	2.	Oil through furnace.
	3.	Steam used for atomization, if any.
B.	Temperatures:
	1.	Oil to and from various sections of furnace.
	2.	Flue gas from radiant section, determined by high velocity couple traverse, reading being taken about every two feet across furnace.
	3.	Flue gas from convection section or after bank of convection tubes through which combustion gases first pass, determined by high velocity couple traverse.
	4.	Air, both atmospheric and preheated, if any.
	5.	Fuel.
	6.	Surface of furnace for estimation of radiant losses.
	7.	Steam used for atomization, if any.
C.	Analysis:
	1.	Fuel, heating value and proximate analysis.
	2.	Flue gas from radiant section by Orsat analysis of samples drawn through water-cooled sampling tube at intervals comparable to those used in temperature measurements.
	3.	Flue gas from same convection section section bank after which temperature (3) was measured, sample taken and analized as above.
D.	Humidity of Atmospheric Air.

METHOD OF CALCULATION

Due to the fact that in petroleum heaters it is often difficult to determine accurately the heat absorbed by the oil in the radiant section from the oil side because of unknown heat of reaction, vaporization, etc., it is essential in most cases to obtain this figure by heat balance from the flue gas side, determining also by heat balance, as mentioned in the paper, any direct radiation from the combustion box to the convection section. Thus, the balance may be shown as follows:

Heat Input:
	1.	Heat of combustion of fuel.
	2.	Heat in air used for combustion.
	3.	Heat in steam used for atomization, if any.
	4.	Heat in recirculated flue gases, if any.

Heat Output:
	1.	Heat in flue gases leaving radiant section.
	2.	Heat to oil in radiant section (determined by difference).
	3.	Heat to external losses.
	4.	Heat radiated from combustion box to convection section (determined by heat balance on part or all of convection section).