The Rosin Equation a n d Combustion Intensity in Gaseous Flumes
ecently a derivation and an interesting discussion of the Rosin combustion intensity equation,
R
I
I=
B/Q(1 $. E/1OO)(Tf/To)tb
has been presented by Essenhigh (1).
(1)
I n this equation,
I is the maximum possible intensity in Btu/ft3/hr; B is the heat of combustion of the fuel; Q is the stoichiometric fuel/air ratio; E is the per cent excess air; T f and Toare the flame and input temperatures, respectively; and tb is the average time required for combustion of an element of fuel. Essenhigh applied the equation to the burning of liquids and solids. This note offers some suggestions about its application to gaseous flames. T h e problem is to formulate suitable expressions for tb. The other parameters present no great difficulty. For an unenclosed laminar diffusion flame, a burning time can be approximated by a method due to Jost (2). If we assume it to apply approximately to an enclosed flame, so that E can be defined, then it is readily shown that
tb
d2/8 D
x
d3/V,
t,
(2)
where d is the diameter of the fuel inlet and D is an average diffusion coefficient. This t, is on the order of one second, in a typical case, and I is small. Introduction of turbulence reduces the burning time. I n a typical unenclosed turbulent diffusion flame, the L / d = constant = 200. Here, L is the length of the flame, and d is again the diameter of the fuel inlet. One can then show that
tb = const
where 6, is the reaction zone thickness and S, is the flame propagation velocity relative to cold gas. Further elucidation of the parameter, 6,) requires the application of a theory of flame propagation. However, one can say that for a typical hydrocarbon flame at one atmosphere pressure, 6, may be about IOM2cm. One finds a tb of about sec and a very high intensity, in the vicinity of l o 9 Btu/ft8/hr. This figure is comparable with Mullins’ theoretical value quoted by Essenhigh in his Table I. A higher figure might be attained in a detonation, where t, might be sec and Z N lolo. This would appear to represent an upper limit to an intensity attainable by normal chemical processes. It should be noted that the average intensity attainable in premixed flames is limited by the total volume of the flame, which may be many times that of the reaction zone. I t is possible to show that, for the (conical) flame as a whole,
(3)
where V F is the volume flow rate of the fuel. The constant is of the order of 100. Aside from yielding higher values of I than the laminar case, this equation also shows that for fixed V,, one will achieve high intensities most readily by using many small fuel inlets rather than a single large one. I n the limit of extreme turbulence, one has the perfectly stirred reactor, in which the intensity approaches that in the reaction zone of a premixed flame. In the reaction zone of a premixed flame, the burning time is readily shown (2) to be
d/G
s,,
(5)
where d is the burner diameter. For typical burners, this tb will be between 10 and 1000 times the tb defined by Equation 4,leading to a typical intensity of about lo7 Btu/ft3/hr, as quoted by Essenhigh. Although, as Essenhigh points out, the Rosin equation is very approximate, its form is such that one can gain some helpful semiquantitative insight into the available combustion intensity, particularly by examining the average burning time. Even in the case of gases, where to is difficult to define, one may find the Rosin equation useful. REFERENCES (1) Essenhigh R H IND.ENOC H E M (7) . ~ ~52 (1967). (2) Gaydon, G.,”Wolfhard: H. G., “Fl~mes,” Chapman and Hall, London, 1960.
A.
HOWARD B. PALMER, PROFESSOR
Defiartment of Fuel Science The Pennsylvania State University University Park, Pa. 16802 VO1. 6 0
NO. 3
MARCH 1 9 6 8
79
