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stirredreact

stirredreact - MECH 6120 Combustion Coupling Kinetics with...

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MECH 6120: Combustion Coupling Kinetics with Thermodynamics At this point in the course, we have covered the topics of thermo- dynamics and chemical kinetics – which are both fairly distinct topics (that is, one is not derived from the other). Our next item of business will be to couple these two topics together to predict the performance of reactors. In a sense, what we will do here is analogous to one of the first topics covered in heat transfer, e.g., the ‘lumped capacity’ solution for transient cooling of an object. To review, say we have a control mass subject to transient heat transfer. The first law, written on a rate basis, would be dU dt = ˙ Q (1) Simply put: the rate at which the internal energy changes is equal to the rate of heat transfer to/from the system (no work here). The above statement is fundamentally sound and complete – no approximations. We can, however, develop a more descriptive prediction of the system by making some assumptions, in particular 1. The temperature of the system, at any point in time, is nearly uniform so that dU = mdu = mc v dT = ρV c v dT . 2. The rate of heat transfer is given by the convective rate law: ˙ Q = - hA ( T s - T ), where T s is the surface temperature, h is the heat transfer coefficient, and A the surface area. Since we assume the temperature of the system is uniform, then T s = T . Putting these two assumptions together gives the familiar first–order cooling law, ρc v V dT dt = - hA ( T - T ) which we can solve easily for the temperature as a function of time. What was the rationale for assuming that the temperature was uni- form throughout the system? You might recall that this assumption implies that the resistance to heat transfer within the system is sig- nificantly smaller than the resistance to heat transfer from the system. Indeed, we could define the ratio of these resistances in a quantity known as the Biot number, and our solution would be valid providing that the Biot number was small. We want to do something similar here – except expanded to include chemical reactions within the system. Kinetics has provided us with rate laws for chemical reactions, analogous to how Newton’s law of cooling (convection) gave us a rate law for heat transfer. And, as was done above, we want to make a similar assumption regarding the state of the system as it evolves in time. In particular, we will assume that the properties of the system ( T , P , mass fractions of chemical species) are uniform throughout the system at any point in time. The validity of this assumption follows a criterion similar to that for the simple heat transfer problem, in that it assumes that transport (heat, mass, momen- tum) within the system is significantly faster than transport to/from the system. This assumption is known as the well–stirred (or well–mixed) approximation.

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