2.25 Problem Sets - Section 5
Problem 5.19: Spark ignited spherical combustion
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A combustible mixture of air and fuel is initially at rest at a density r1 and uniform pressure p∞. At t = 0, the mixture is ignited at the origin by a spark and a flame front begins to move radially outward from the origin at a constant speed Vf. As the flame front overtakes a gas particle, it converts the combustible mixture of density r1 to a hotter combustion product of lower density r2. This occurs so fast that the flame front can be modeled as an infinitesimally thin density discontinuity. Because the volume of a fluid particle increases as it is enveloped by the flame front, the combustible mixture ahead of the expanding flame front is pushed radially outward by the front. The combustion products of density r2 that are left behind the flame must, however, be stationary relative to the reference frame fixed in the origin.
(a) In a reference frame fixed in the origin,
determine the radial outflow velocity v(r,t) of the combustible
mixture. There will be different expressions for the unburned gas
ahead of the flame front (r > Vft) and the
burned gas left behind (r < Vft). See also
Prob. 3.17. Show that the velocity distribution can be expressed
in the dimensionless form 
, where
is
either a dimensionless radial distance at fixed time or a dimensionless
inverse time at fixed r, and 
is
a dimensionless density difference between the unburned and burned
gases.
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(b) In a real combustion problem, the quantity that would be specified, as a property of the combustible mixture, would be not the flame speed Vf relative to the origin, but the flame speed Uf relative to the fluid just ahead of it. Show that in this problem the flame front does move at a constant speed relative to the fluid ahead of it if Vf is constant, and find an expression for Vf in terms of Uf and the given densities. |
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(c) Assuming gravitational and viscous
effects are negligible, write an equation for the pressure
gradient at r, t in the combustible mixture ahead of
the flame (r >Vft) and determine the
pressure distribution p(r,t) -p∞ there.
Express the pressure distribution in the dimensionless form
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(d) Show that there must be a pressure jump across the flame front, the pressure on the side of the combustion products being lower than in the mixture just ahead of the front. Derive an expression for this pressure discontinuity. (Hint: Use the control volume shown in Fig. 2.) |
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Can you hive a physical explanation of why the pressure is lower behind the flame front than ahead of it? This seems not to make sense at first sight: how can a lower pressure region accelerate a higher pressure region radially outward? (e) For g = 0.25, say, plot the function P=f(s,g) from s>=0 to s=10. Also show the limiting cases g = 0 and 1. |
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