 mainly the combustion problem now so combustion problem of essentially having these species conservation equation equations rather and energy equation course energy conservation equation so this is what we want to now focus our attention on we simply say flow field is prescribed and let us not worry about the two way interaction where we are assuming a density dependent temperature that influences the flow back again the moment we will just say as far as the combustion problem is concerned the flow field is prescribed and the question is is it possible for us to now simplify this set of equations right what do you mean by simplify how do you how do you want to simplify this well if you think about it the species conservation equations is something which contains the V the diffusion velocity where the species diffusion velocities which are in turn given by the multi-component diffusion equation so you now have these n equations relating to three n equations for the three components of this multi-component diffusion vector equation okay and that those are those are an xis whereas this is in yis you know have to have an additional a set of n equations to relate xis and yis so that is complicated enough all right and then you have this energy equation which has like lots of terms that are on the right hand side and each of those terms has lots of terms in turn and so on so it is actually a very long equation our goal believe it or not okay at this stage is to try to simplify these two sets of equations to look almost the same not even similar is that possible can we can we even dream up of making this set of equations and a soup ball of additional equations that it comes along with to look like an energy equation that is very long actually so it is possible if you now make lots of assumptions right but the goal here is goal to simplify the above set of equations to reduce to a common form reduced to a common form in fact let me tell you what the form that we are looking for is say a operator L script L of something like an alpha I okay equal to omega and alpha so this is I equals 1 to n right so obviously that is a species equations right species equations then let us suppose that we can now form a alpha T which is also equal to omega okay and this is the energy equation right and that is one equation what we expect is alpha I should be related to why I in some way okay the primary unknown corresponding to the species equations why I being the mass fraction of species I and alpha T should be related to temperature in some sense give you know in a fairly direct way so long as we can actually establish those kinds of relationships and then say this stands for temperature that stands for species concentration then and then you now are able to write these two equations in this form okay then we can now form form then we can form we can form beta I equals alpha I – alpha 1 you pick the first one okay first of the alphas okay and then subtract that from all of their alphas and then you form a beta I corresponding to that okay and beta T equals alpha T – alpha 1 you pick the first of these alphas that is okay all right and then subtract from alpha T to form a beta T okay now look for L the script L to be a linear operator okay further now if you really think about it if you know if you know some basic fluid mechanics what you would expect is the nonlinear terms in fluid mechanics are primarily convect coming from the convective term in the momentum equation okay if you are prescribing the velocity field in the species conservation and the energy conservation equations the non the convective terms in those species equations and energy equation are not not they are not nonlinear because your velocities there are known the momentum equation you have a product of velocity times velocity derivatives and the velocity was an unknown any time you have a product of an unknown and it is with itself or its derivatives then that shows a person nonlinear term but in the species equation and the energy equation if your flow field is prescribed that constitutes a linear term with a variable coefficient or a non constant coefficient that is all okay so as far as the convective term is concerned which typically poses the nonlinearity in fluid mechanics in the momentum conservation as far as the combustion problem is concerned we can expect it to be linear that is fine all right it is the other nonlinearities that we will have to worry about like for example if you had a radiation term with the T to the power 4 or if you had this e to the – EA over RUT right the exponential term of the reaction rate depending upon temperature and so on those are the villains that we should be looking out for but we expect that that should actually be on the right hand side okay the L itself could be linear so if you now look for this L to be a linear operator then we get L of alpha 1 equal to omega but L of beta i equal to 0 and L of beta t equal to 0 here I going from 2 to n because you can now subtract one equation from the other right you all you can try basically subtract the first equation from all of the equations and then now form your pairs of alpha i – alpha 1 or as for t – alpha 1 and then plug in the betas the right hand side is the same for all of them so if you now subtract one from the other you get a 0 right what is the advantage with zeros I really we really like zeros do not we right so the moment you now have zeros the first of all it means that this really means that we are now looking at homogeneous equations right which are a lot easier to think about when mathematically speaking when compared to inhomogeneous equations where you have source terms influencing the mathematical nature of the equations otherwise okay so first of all you are dealing with homogeneous equations that is like a big boon but implicitly we are also thinking it does not have to be zeros but thank God the nonlinear term goes away it was the right hand side that was nonlinear we had this e to the – e exponential of – e by rt which is a huge nonlinear term I try to get rid of this in most of the equations that was posing this stiff problem right that was the one that was actually trying to vary in a highly nonlinear fashion that means it becomes very sensitive within a very narrow range of temperatures and you have to take like very very small steps in time or space or both in order to try to capture that variation somewhere there near the flame and so on all those things you do not have to worry about for these equations you still have that stuck in one equation representing this for the representing the severity for the rest of the set and if you were to follow this then here we would we call beta i and beta t as coupling functions because they couple alphas together okay beta i is couple alpha i and alpha 1 and beta t couples alpha t and alpha 1 right so these are called coupling functions. So our goal is essentially to simplify the above set of equations to reduce to a common form that is L of alpha is equal to 0 omega for both the species and the energy together and then form the coupling functions so that we could actually get a homogeneous set of equations for a whole now number of large number of equations except just one okay sounds very incredible at this stage if you now think back on what the equations are but the question is how do we simplify so this is the starting point right so we will now have to make a lot of assumptions 11 assumptions to be precise okay which are mostly simplifying assumptions so we now make these assumptions and I want you to listen to this very carefully there are 11 assumptions that we are going to make all right so it could be as if you can you can expect an exam question or a PhD qualifiers question what are the 11 assumptions of the Schwab-Zeldovich formulation because they are the 11 assumptions okay so we will make those 11 assumptions number one number two whenever we make assumptions keep in mind we get into the habit of questioning those assumptions are they good assumptions or are they bad assumptions are we making them mainly because we want to simplify the matter right or they just simplifying assumptions or are they physically tenable that means or we can be justify them from physical grounds are they reasonable right we will have to go through that when we make those 11 assumptions the third is there a way we can relax these assumptions some of them many of them all of them right so those will be your homework problems right so I will try to point out to you which are the assumptions you can you can relax and has been relaxed in the literature okay and I will also point out to those literature and then you can you can say well work out the Schwab-Zeldovich formulation with this assumption relaxed that is like one particular question in an exam okay so if you thought that you really mugged it up then the next question in the next example be work out the Schwab-Zeldovich formulation with relaxing the other assumption so you can keep doing this okay and it will be it will be a pretty good piece of work actually okay so do not take it lightly it is very important so I am not going to relax all the assumptions I am not that crazy okay so I will point out to some assumptions that can be relaxed so the first assumption that you are looking for is negligible body forces we also should be looking for why we are making these assumptions right so and then I will also point out to some things that we are not assuming as you go along so in that sense to do to that extent the formulation is general enough because we are not making assumptions for example to tell you right away we do not have to make assumptions about constant CP's we could have temperature dependent species that are different for different species all right that is that is great that means you can actually get the diabetic flame temperatures predicted fairly well and then your formulation should be able to show the temperature in your flow field so that is pretty good here the reason why we want to actually neglect body forces is it shows up in your it shows up in this new set of equations it shows up in the multi-component diffusion equation right so our goal is to try to simplify the multi-component diffusion equation to look like fixed law all right so that we can directly substitute the fixed law form of the multi-component diffusion equation into the species conservation equation right away just like how we did it the first time for a binary mixture then we do not have to deal with the multi-component diffusion equation separately anymore so the question is what will allow us to simplify the multi-component diffusion equation to look like fixed law you remember the multi-component diffusion equation had four terms the first of those was the one that was going to give rise to fixed law then there was a pressure gradient term then there was a temperature gradient term which was the sorate effect and then there was a body force term so if you wanted to get rid of pressure gradient term and then you have to get rid of the body forces term then you have to get rid of the sorate effect okay so the first thing that we are trying to do here is to neglect body forces now if you get went back and actually looked at that particular term you had a fi vector – fj vector so you do not necessarily have to neglect body forces altogether it was sufficient if the body forces were the same on all the species right so if not if not okay equal on all species that is assumption that we try to make okay second negligible sorate and due for effects terms involving dt, I right this is this is the sorate effect is actually neglected so that you can simplify our multi-component diffusion equation into a fixed law the due for effect actually shows up in the energy equation which also we tried to neglect primarily the justification being dti over rho dij is typically fairly small it has one or magnitude less than unity all right therefore it is insignificant or negligible most of the time pressure gradient diffusion pressure gradient I should say negligible pressure gradient diffusion right negligible pressure gradient diffusion this term was looking more like yi – xi times grad p over p there are many ways by which you can neglect this one you simply say this term does not contribute a lot to the multi-component diffusion equation right no matter how much it is you say it is very you know evaluate it and find that it is very small the other way of trying to get rid of the get rid of this term is to say that you have equal molecular weights that means yi is equal to xi but that is a worse assumption than just saying that this is negligible right the third way of actually trying to justify this is just the previous class we found that for low Mach numbers the pressure is almost a constant it affects only your flow field but it is not really get into your combustion field a lot right so as far as the combustion problem is concerned we could simply say the grad p is very small right that is the reason why we want to say the diffusion associated with it is even smaller right okay. So that is reasonable now neglect negligible bulk viscosity bulk viscosity and viscous dissipation these are related but slightly different bulk viscosity effect can be neglected for incompressible flows when you are now looking at low Mach numbers we are now talking about low incompressible flows so you can say that the divergence is equal to 0 so you do not really have dilatation effects coming in so bulk viscosity the effect of bulk viscosity is negligible and this also shows up in the viscous dissipation term and we are now saying the viscous dissipation itself is pretty small if you now do an order of magnitude analysis unless you are looking at situations of tribology like when you now have a two surfaces that are actually in very close clearances and you have a fluid in between like in lubrication and so on or if you have like reentry effects where your Mach numbers are so large and the boundary layer is so thin with a steep gradient effectively you are looking for very large gradients of velocity that is when viscous dissipation is significant otherwise you do not have to worry about it not typically in our in our application 5 negligible radiant heat flux right qr qr vector right now of course this is this is going to be a non-linear term if it were present so you are going to have a t to the power 4 and t is an unknown so you are now looking at a non-linear effect there so try to get to the fit okay but of course when can you when can when can you say this is okay so long as you are dealing with a gaseous medium this is reasonable okay but the moment you have things like surfaces you have to think a little bit harder so if you have surfaces you have to look for whether those surfaces are transparent or opaque or if they are closer to black if their emissivities are closer to back blackbody emissivities right so if that is the case then you cannot neglect this so not valid not valid when in problems with phase interfaces right not valid in problems with phase interfaces well of course you will find in the literature there are lots of problems with phase interfaces that are being solved after neglecting radiant heat flux and you still get good results it is on it is not terribly bad okay but in general the radiant heat flux is significantly small think about this you now look at a flame the fact that you are able to see the flame is because you have radiation coming and hitting you your eye all right and you are like wait a minute I could I could see the flame only because of radiation it is there how could I say how could I not take that into account the answer is that is quite small okay in magnitude so you do not have to worry too much about this 6 we assume a steady flow we did that when we were trying to do on the low Mach number assumption okay so whatever we did the in the previous class where we came to the conclusion the pressure remains almost a constant for low Mach numbers is typically called the low Mach number assumption right so the low Mach number assumption was actually derived for steady flows which is something that we also adopt over here the point I mentioned the other day was you do not have to do that okay and similarly we do not have to do that here is here as well you can rely this assumption can be relaxed you should be able to relax this assumption and derive a unsteady equivalent or an unsteady Schwab-Zeldovich formulation okay that is a homework problem for you and keep in mind what are we looking for we are looking for a L the script L operator right that should be linear and typically the time derivative dou by dou t is a linear term it is not showing up any non-linearity so you should not have a problem with unsteady flows at all so you should be able to relax this assumption. Then we say binary diffusion coefficients coefficients of all pairs of PG right or equal that is there is DIJ equals DIJ equals D not even saying that we have DIJ is equal to DI that is to say we are not saying that the binary diffusion coefficients are the same as the multi-component diffusion coefficients recall the multi-component diffusion coefficient coefficients are derived quantities whereas the binary diffusion coefficients are more fundamental okay which come from kinetic theory but we are saying here that all of them are going to be equal for all pairs of species and this is very important for you to be able to plug in your multi-component diffusion equation into a fixed law and then plug that into the species conservation equation otherwise what happens is you now have this XI is equal to sorry grad XI is equal to some coefficient times VI minus VJ and you could derive fixed law only for a binary speed binary mixture where again DAB is equal to DBA and since only two of them are there and they are equal it is just as good as this assumption and then you could have you could have obtained fixed law that is what we did before but here we have to get past that for a truly multi-component mixture more than two species therefore this assumption is necessitated okay the question is is it okay the answer is not terribly bad typically you have to start worrying about this assumption if you are dealing with things like hydrogen as part of your mixture right let us say you had a mixture of hydrogen and hydrocarbon going through combustion then the pairs of species involving hydrogen may typically have a higher diffusivity when compared to others and then that is markedly different whereas if you are dealing with most species of the order of the same molecular weights not necessarily equal you can expect that this assumption is a very reasonable assumption not not bad at all okay 8 low speed or low Mach number low Mach number this implies that momentum equation reduces to P is equal to constant to first order which means we could now say let me get a flow field that is prescribed I do not have to worry about solving for the flow field in addition to the combustion problem so I am looking only at the combustion problem so that is something that came came from the previous class 9 unity lowest number right so question is what is Lewis number right so Lewis number le is k is a k divided by rho Cp d now keep in mind that d should really be like dij in reality alright so correspondingly you should have a Lewis number that is like le ij okay but many times as we have seen earlier the dij can be replaced by like a di that is like the multi-component diffusion coefficient rather than the binary diffusion coefficient in which case again we will now have to recognize that le could be written should be written as a lei that means it is different for different species so you are looking at a Lewis number that should be really species dependent and many times people talk about a fuel Lewis number or an oxidizer Lewis number that are distinct from each other right so what is this Lewis number really telling us k over rho Cp is thermal diffusivity and dij or d here is the mass diffusivity okay so in si units k over rho Cp is going to be like meter squared per second and the mass diffusivity also has the same units meter squared per second and therefore le is a non-dimensional number which actually indicates the capability of a species to conduct heat just as well as it can diffuse right so we are now looking at comparing the capability of the species to actually transport enthalpy as it can transport mass right so that they do not have to be necessarily the same right depends on how well it can conduct versus what its Cp is in comparison to what would how much it can diffuse itself as a mass right why would you want to now compare these two and what you are now basically saying is this should be equal to 1 this is a very very very important assumption it can be relaxed that is a good news okay but it is a very important assumption why because our goal is to actually look at the species conservation equation and the energy conservation equation and try to express them in a single form how do you hope to do that if you know you do not expect the species to actually transport enthalpy just as well as it can transport the mass because in your these equations are essentially remember the convection diffusion reaction these are the three things and if you had some sort of an intuition you would say that omega stands for reaction not just because it is a symbol but I have been saying that this is the one that is containing the nonlinearity in the Arrhenius term that is essentially from the reaction if you now say convection diffusion and reaction and this is going to contain the reaction then this is going to contain the convection and the diffusion and sure enough prescribing a velocity you now simply have a non-constant coefficient for the convective term coming from the flow field and the diffusion term should both should be linear just as well as the convection term. So we expect this to turn out to be linear that means we are looking for two terms here both of them linear one of them coming from convection the other one coming from diffusion right if now these two terms should look similar to similar for the species conservation equation and the energy conservation equation then we should be looking for the transport terms to be comparable. So if you now have a species which is actually diffusing so fast but it does not really conduct heat a lot that means it is having a non-unity Lewis number less than 1 okay then the the transport term so the or the diffusion terms in the species conservation equation and the energy conservation equation they are not going to match up all right you are going to have a problem. So the only way we can actually try to get these two together to look similar is if you now have a unity Lewis number in fact in most of the earlier literature I will I will say too early like as late as even the 70s or 80s or even now some some some some some works or many theoretical works that are getting published they would make a unity Lewis number assumption and sometimes when they relax this assumption and publish they would even put it in the title like they would say blah blah blah blah for non-unity Lewis numbers it does not like it is a big achievement that they have done that they relaxed this assumption right so that that indicates the importance of this assumption okay that is indicating the importance of this process of what is called as preferential diffusion when you now say the Lewis numbers are first of all not equal to 1 and not equal to each other so if you now say preferential diffusion you now allow for a oxidizer Lewis number to be different from a fuel Lewis number for example right and sometimes you have this analysis that is done where you now assume that the oxidizer Lewis number is unity but the fuel Lewis number is not unity or vice-versa maybe okay so many times you need this this assumption in different ways but it is still possible for us to think of writing not exactly in this form of course but you will still be able to write something like a convection diffusion and reaction okay except that the diffusion terms alone may be different by a factor of the Lewis number if the Lewis number were non-unity and that would be like the counterpart of the Schwab-Zelovich formulation we may not be able to form coupling terms alright so it is possible to actually work with non-unity Lewis numbers and proceed in the manner if you now make all the other assumptions relax this assumption and see what happens it is it is worth pursuing that okay then we assume a chemical changes occur in a single reaction step single reaction step sigma i equals 1 to n nu i single prime script mi gives rise to sigma i equals 1 to n nu i double prime script mi just one step we do not entertain multi step chemical kinetics okay that helps us in dealing with trying to split the enthalpy into the formation enthalpy and the sensible enthalpy and deal with the formation enthalpy term so you now get a single heat release rate term because of the single step kinetics it is essentially what is happening is you now assume like there is only one reaction that is happening in the flame although there are lots of reactions that are happening but if you were to entertain many reactions each of those reactions is going to give rise to a heat release right and some of them could actually absorb heat so there is like a net heat release that you should be looking for whereas we want to deal with only one heat release rate term not multiple heat release rate terms okay so that that is what this assumption lets you do try relaxing this either okay so we will talk a little bit about that as we as we go along the 11th assumption is a fairly straightforward assumption no external heat flux that is Q dot equal to 0 Q dot is equal to 0 this was showing up as a solitary term only in the energy equation it was not there in the species equation so if you now keep this then it is not possible for you to make them look alike right so we have to get rid of this this is reasonable again okay you get into this kind of situation only in some odd ball cases where you have to keep some externally heat generating mechanism or machinery within your domain alright that is like some electrical heating or nuclear heating or let us say arc heating or something of that sort which is not which is not due to the combustion alright therefore we do not have to worry about this under most circumstances good so we now went through the 11 assumptions now we go through the algebra how do you crank the machinery to make make this look like we make make these two equations look alike so it is going to involve some magic alright so you have to kind of watch out for how these manipulations happen and we will go through this now and maybe in the next next hour as well and maybe by the end of then we should be we should be getting there to see whatever we are looking for yeah yeah the equations can still be considered linear if you now have like two species equations you assumed a non-unit of Lewis number alright but the Lewis numbers were equal right you now have two species equations which are still any which can be where you can still form the coupling functions alright and then you have the Lewis number the equal Lewis number for each of those still sticking together it is only that you cannot look make it look similar to the energy equation so as far as the species equations are concerned they can begin to look similar right and I am still saying I am not having a unity Lewis number I am having non-unity Lewis number but non-unity but equal you see so there are such conditions there are also other situations where it is possible for you to deal with like saying one of the Lewis numbers is equal to one but the other other Lewis number is not equal to one there you try to couple that equation with the energy equation where the Lewis number of that particular species is equal to one but you keep the other one okay so typically we will in fact what we will find is in the problems that we deal with let us say we want to do like a premixed flame or a diffusion flame and so on in your course our starting point will be the Schwab-Zeldovich formulation okay at that time we should look forward to seeing if we can relax the Lewis number unity assumption for some of the species or say equal Lewis numbers but not necessarily equal to one and so on the point I want to make was what you will find us we will pick and choose some equations to form coupling functions and not worry about the others all right so this will depend on the way you assume your Lewis numbers all right so it is it is it is it is it is pretty interesting the way we can deal with it okay so now given the situation we now start with the overall continuity equation it simply leads to divergence of rho v vector is equal to 0 right we still deal with 3D we will have a vector form of equations so that means it is a three-dimensional formulation nothing no assumptions on that right now let us look at energy equation with all the assumptions that we have made so no q dot viscous dissipation phi and dp over dt you get this dp over dt only for compressible flows because we are now looking at a low Mach number situation you do not you should not have to worry about this then you have a rho v vector dot del H is equal to del dot k grad T minus del dot rho sigma capital K equals 1 to n H k yk capital Bk okay this is coming from your divergence q vector and q vector had k grad T my okay kk grad T in fact minus k grad T and you had this this term and you had the due for effect and the qr you have we have got rid of the duty due for effect in the qr outside of the grad q dot you had the body force work the you had the viscous dissipation and you had the pressure work so all those all those have gone so we had left with only these two keep in mind what are we looking for we are looking for getting this equation to look like convection diffusion reaction now the reaction rate term is actually embedded in here this is this is going to be now written as h is equal to sigma yi sigma sigma yk hk where hk is the specific enthalpy of species k which can in turn be written as delta hfk not plus the sensible enthalpy for that for that right and then the delta hfk not term is the one that is going to contribute to the heat release so that is where the reaction term is coming from in the energy equation okay the job of the reactions in the energy equation is to give rise to the chemical heat release there is embedded in here right so this contains both the convection term and the reaction term that we are looking for this is the diffusion term that we are looking for from an energy point of view this is the this is essentially the energy transport term okay that is something that we have to deal with it is not something that we were looking for it is not there in the species conservation equation okay but you do have a capital vk showing up in the species conservation which will now have to be replaced by a fixed law having a gradient yk so this is going to be a little bit more involved than then it seems okay so now note that h is equal to sigma h yk yk hk all right now therefore sigma yk hk divergence rho v vector equal to 0 starting from here I just multiplied these two right and where inside of this is equal to 0 therefore this should be equal to 0 now add the above to the energy equation and then you have so what do you get if I can now add this to this which is essentially 0 okay I can now try to take the divergence out you see you have a rho v dot grad h here we have a this is h times divergence rho v so if you now add these two together I can get a divergence of rho v times the h you see so I can get this divergence of rho sigma hk yk and then what happens why did I do want to do that because I have a divergence of rho times this right I should now be able to combine these two v plus capital vk right and I also note that this is like a divergence term and so if you now have a divergence that is outside I should now be able to write this also inside k gradient T equal to 0 now let us call this a keep this why would I want to do this because I am now going to begin to realize that hk can be written as ? hfk0 plus the sensible enthalpy all right ? hfk0 times yk times v if you now have this term I can now try to look for supplying the species conservation equation to get its right hand side to come and plug in here okay so I will now be able to write something like a the for the for the term that has the ? hfk0 alone I should now be able to write this is like a ? wk ? hfk0 so wk is coming from the right hand side of the species conservation which will now look which whose left hand side is this I plug the right hand side of that here right for only that term which has the ? hfk0 that will simply mean the chemical heat release rate term is nothing but the reaction rate term times the heat release from that reaction that is that is correct because a heat a reaction is going to give rise to a certain heat release per unit mass of the fuel or any of the reactants if you now multiply that by the rate of change of that species mass per unit time right then you should get the heat release the heat release rate so we will try to look for that manipulation soon so we now say species conservation can be written as divergence ? yi times V plus capital VI equals Wi equals 1 to n right now let us just stop this saying this is B okay so what I am basically saying and I will stop stop so basically note that so note that hk equals ? hfk0 plus integral t ref to t cpk small cpk dt right so this is the this is the sensible enthalpy term this is the standard heat of formation therefore if we now plot this in here right in the energy equation and notice that you now have a ? yi V plus Vi combination with the ? hfk wherever you have that you now put your Wi well I guess I am sorry I missed symbols so let us just have k instead of i capital K right so what you are looking for is whenever you have this combination you now try to put wk whenever you have ? hfk0 over there and if you are not split this then what is going to remain is only this term on the left hand side the right hand side is simply going to be ? hfk0 times wk sigma of that so the energy equation energy equation becomes becomes divergence of divergence of rho integral on rho we now say sigma well actually I think I can get rid of the sigma for the first term if I now try to pull this V out and open up this brackets sigma hk yk will simply again be H and that will correspond to simply notice that we do not have a k over here because we summed over there so we are now looking at only the sensible enthalpy term kept on the left hand side so we now sum over the Cpk becomes Cp but then you have to keep the Cpk as it is with the capital Vk so we will we will just do that sigma k equals 1 to n yk capital Vk and then keep your sensible enthalpy term as it is with the Cpk dt and then of course we also have this minus k dt grad t that you have to keep minus k grad t and what we did was to keep get the standard heat of formation part to the right hand side and notice that that part alone actually stands for wk so you can now easily write this as minus sigma k equals 1 to n delta H of not k wk so this obviously is the chemical heat release term right term because this actually tells you how much is the heat that is released right so this is actually like joules per kg and this is like kgs per meter cube second right so if you now get rid of these two kkg this is the amount of heat released per unit mass of that species k and this is the mass of species k that is getting converted per unit time per unit volume so together this will actually mean joules per second per meter cube or watts per meter cube you see so this is the volumetric heat release rate right so that is how and then you sum it sum it over for all the species that are getting produced or consumed okay so this effectively is now for the first time you are now beginning to see the chemical heat release rate isolated in the energy equation yeah okay this is a good point to stop.