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Show such that the Arrhenius law (eq. (17) for the hydrocarbon reaction rate can be written as: RArrh.CMt ~ ^Arrh.CJH) e RT (22) To obtain the same reaction rates as the EBU-model predicts, the Arrhenius rate has to fulfil the following equation £ REBU.CM^ = AmixP fr^CxH) E„ (23) = C Arrh. CM e RT =R ' ' ff Arrh.CM. ff Replacing Ea by eq. (19) yields the following equation for RT C,(^.^....) = l + ln AmixP £ ^CMt "Arrh. CM' » ff J (24) Since the right side of eq. (24) contains no unknowns, C, can be calculated and added to the field array of all data sampled using condition eq. (20). A multiple linear regression ( M L R ) analysis of the corresponding flow properties and the calculated C, values returned the following algebraic function: P-k YCxHA (25) with C ^ = 2.395 (0.2333), a = 0.34 (0.0279) and b - -0.734 (0.0497). The values in brackets are approximated standard deviations to the solution of the multiple linear regression analysis. The overall quality of the solution given in percentage of the standard deviation is 86%. The "novel" finite rate expression m a y be written as E.\ I-O ^ <:.">)V) R ,CMV ~ C Arrh.CM „ RT (26) Because the above Arrhenius-type reaction rate includes a "turbulent" term and represents an expression generally used to describe chemical "kinetics", the model is given the name "turbulent-kinetic" combustion model. In the following it will be abbreviated as TK-model. Applying the derived constant CE a = 2.395, a = 0.34 and b = -0.734 it is expected that, close to flame temperatures of 1200K, reaction rates by means of eq. (26) in general overpredict the E B U reaction rates. This is due to the fact that weighting factor, which would be preferable to minimise the statistical impact of the high EBU-reaction rates at temperatures lower than 1000K, cannot be introduced to the M L R analysis. However, a parameter study on the coefficients CEa, a and b shows that when CE a = 2.5, a = 0.25 and b = -0.75, the finite rate predictions match the reaction rate zone as shown m Figure 10 reasonably well. The changes to C E a and the exponent b are m the order of 4.2%, respectively 2.1%. The exponent a is altered from 0.34 down to 0.25. A discussion of the physical meaning of eq. (25) m a y help to justify the proposed change of the parameter a by 3 6 % . The basis to the exponent a is written as £ v k ' v (27) and hence proportional to the E B U reaction rates of C xHy (eq. (4)) £ £ P~k~YC,H) * REBU.CMt = AmixPTYCMy (28) The functional relationship between the effective activation energy Exc„ (eq. (19)) and the E B U combustion rate says that if the turbulence driven transport through the eddy-break- up regime is high, finite rates are increased as well. The rates are scaled such that In [R TK.CM. *[R y EBU.CxHyJ (29) represents an exponential function which clearly indicates that the influence of the turbulent "eddy-break-up"- mechanism on this quasi-global finite rate exists, but is damped by the power a. The originally derived exponent of 0.34 describes a weaker damping. This weaker damping should be avoided because the high local reaction rates at temperatures lower than 1000K, as predicted by the E B U - model, are known to be unrealistically high. A fit of the parameter a is therefore justified. Discussion of the physical meaning of the TK-model. From a physical pomt of view it m a y be considered that m high turbulence regimes and at low flame temperatures the "eddy-break-up"-driven mass transport, which "channels" unmixed fuel and oxidiser species from large length and time scales down to the smallest scales needed for chemical reaction to occur, is too high due to the significant influence of the turbulence term Amix £ k in the EBU-rate formulation (eq. (4)). The smallest scales which are addressed by the E B U rate formulation may be considered to be in the order of the laminar flame thickness (Peters, (1986)). Ronney and Yakhot (1992) argued that the influence of small scale turbulence structures in high Reynolds number flames are responsible for flame broadening effects. Their findings for premixed flames showed that at scales smaller than the laminar flame thickness turbulent fluctuations exist which increase reaction rates. Hence, it is believed that a portion of the turbulent kinetic energy drives very small scale fluctuation within die laminar flame layer. The physical function of the derived TK-model may be expressed in similar terms: 10 |
