
EVALUATION OF THE EQUIVALENT
SURFACE, aA_{cp}
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.
Hottel^{8} gives a as a
function of the ratio
as in Figure 5.
The following example illustrates the method of calculating
aA_{cp}. 
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 



aA_{cp}=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.
EVALUATION OF THE FLAME
EMISSIVITY, P_{f}
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
Hottel^{9} which were used in this calculation are included in the
Appendix as Figures Nos. 13 and 14.
P_{CO2} 
= 
partial pressure of carbon dioxide in the flue gas, atmospheres 
P_{H2O} 
= 
partial pressure of water vapor in the flue gas, atmospheres 
L 
= 
mean length of radiant beam in the combustion chamber, feet 
PL 
= 
atmospheresfeet 
Values of P_{CO2} + P_{H2O} and the airfuel
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.
MEAN LENGTH
OFRADIANT BEAMS
The mean length, L, of the radiant beam in the combustion chamber may
be estimated from Table I.
Table I.Mean Length of Radiant Beams in 
Various Gas Shapes^{12} 
From the values given by Hottel^{12}
for the mean length, L, of radiant beams 
in various gas shapes, the following
approximate values have been 
derived for use in practical oil heater design.
These values 
may be used until more exact methods are
available. 
Dimensional Ratio 

(length, width, height 
L 
in any order) 

Rectangular Furnaces 

1. 
111 
to 
113 


121 
to 
124 
2/3(Furnace Volume)^{1/3} 
2. 
114 
to 
11inf 
1 x smallest dimension 
3. 
125 
to 
128 
1.3 x smallest dimension 
4. 
133 
to 
1infinf 
1.8 x smallest dimension 
Cylindrical Furnaces 

5. 
d x d 


2/3 x diameter 
6. 
d x 2d 
to 
d x inf 
1 x diameter 
EVALUATION OF THE
OVERALL ECHANGE
FACTOR, f
Let 


A_{t} 
= 
total area of furnace surfaces in the radiant section= 


A_{cp} + refractory surfaces unprotected by tubes, sq.
ft 
aA_{cp} 
= 
equivalent cold plane surface, sq.ft. 
A_{r} 
= 
effective refractory surface, sq.ft. (A_{r} =
A_{t}  aA_{cp}) 
P_{c} 
= 
emissivity of the ultimate heatreceiving surface, assumed 


=0.90 
P_{f} 
= 
emissivity of the flame. 
F_{rc} 
= 
fraction of all the radiation emitted from all the refractory 


in all directions, which, if not absorbed by the gas, 


would hit cold surface aA_{cp} 
The overall exchange factor, f, as defined by
Hottel^{9}, 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 evaluation^{10} 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.
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