MECH E4320 (Fall 2024): Homework #6
1. Resolve the 1-D chambered non-premixed flame problem solved in class but now for a system with PDF ≠ PDo ≠ λ/cp (but still with constant PDF , PDo , and λ/cp throughout the chamber) using only the reaction sheet assumption (since you cannot apply the simple coupling function presented in class to non-equidiffusive systems). Use th~e ~fact that~there are no reactions from 0 ≤ x < xf a~nd fr~om xf <~x ≤ l to define functions~ for~YF , Yo , a~nd T on either side of the reactant sheet (i.e. YF— , Yo— , and T — for 0 ≤ x < xf and YF+ , Yo+ , and T+for xf < x ≤ l ). Then apply the reaction sheet jump relations and continuous function requirements (for species and temperature) (with “no leakage”) to provide additional e~ffec~tive bou~ndary conditions that allow you to solve for the coefficients in the expressions for YF , Yo , and T on either side. (6~0 points)
a. Solve for the flame location,x(~)f, and flame temperature, Tf . Rework them into a form that
allows for a straightforward comparison between the equidiffusive system analyzed in class and the present non-equidiffusive system. (Use the expressions highlighted in red below for comparison.)
b. How does the flame location for non-equidiffusive systems differ from that ofequidiffusive systems? Physically, why would that be?
c. How does the flame temperature for non-equidiffusive systems differ from that of equidiffusive systems? Physically, why would that be?
Note that to find the flame temperature in the solution for equidiffusive chambere~d flam~es presented in class, we ca~n evaluate the temperature at either xf— or x (using T — or T+ , respectively). Evaluating T — at xf— gives:
Separating the second term and using our expression for xf in one of those separated terms:
gives
where the boundary temperature terms represent the sensible enthalpy in the reactants (weighted by proximity of the flame to the boundaries) and the final term represents the chemical enthalpy converted to sensible enthalpy from combustion (as we will see more clearly next class). Use the expressions highlighted above in red for comparison in your answer to question 1.
2. Perform. some initial aspects of an analysis of premixed flames for simple idealized systems. Again, keep in mind that many concepts even from this simple idealized analysis still often apply to the local structure and trends of more complicated, even turbulent, flames, such that many of our results are in fact more general properties of premixed flames. Here, consider a steady-state, adiabatic, laminar, and 1-D planar premixed flame for sufficiently off-stoichiometric systems where one reactant is present in large excess in the pre-mixture, for the reaction
D + E → P
where D is the deficient reactant and E is the excess reactant, which is present in sufficient excess that its mole fraction is roughly constant, i.e. YE(x) ≈ YE ≈ constant, throughout the entire domain. In such a way, it is then only necessary to keep track of variations in the deficient reactant, YD. You can also assume that PDi and λ/cp have constant values throughout the domain but are not necessarily equal to each other, i.e. PDi ≠ PDj ≠ λ/cp .
Laminar premixed flames are (subsonic) waves that propagate through a reactive fluid mixture at a given velocity called the laminar flame speed, su . If we choose a reference frame such that the unburned flow velocity far upstream of the flame, uu, is equal to the laminar flame speed, su, then the flame is stationary in that reference frame. The boundary conditions for such a system can then be expressed as
Where indicates that all derivatives of all orders, k, are equal to zero. The subscripts u
and b are used to refer to the unburned and burned mixtures.
a. Use the continuity equation to show that f is constant throughout the entire domain and that the unburned and burned flame speeds can be related by the density ratio across the flame. Given that ρu/ ρb is usually in the range of ~5-10, is the burned flame speed higher or lower than the unburned flame speed?
b. To relate the burned gas temperature (i.e. the flame temperature) to the upstream boundary conditions, it is useful to consider our earlier equation for total enthalpy, h, instead of our more commonly used equation for sensible enthalpy, hs .
Integrate this equation from -∞ to +∞ and use the boundary conditions above to show that
hu = hb
which is exactly the same equation we used for adiabatic flame temperature for constant- pressure, homogeneous conditions. Then, separate the total enthalpy into sensible enthalpy and enthalpy of formation for the mixture, h = cpT + Σ Yihi(o) , to show that the equation above becomes
where qc is the heat of combustion per unit mass of the deficient reactant.
c. If the reaction rate term can be described by
where we have used v’D = 1, and defined a new expression for Bc (that you do not need to show!)
Then, show that our familiar equations
can be transformed into the non-dimensionalized equations
For which a non-dimensionalized x is defined as
where (λ/Cp)/fo conceptually represents a characteristic flame thickness (where the symbol f o is used instead off to emphasize that it is constant throughout the domain), and
is the collision Damkohler number comparing characteristic collisional times (of reaction) and characteristic transport times.