 Okay, in the last segment we took a look at a very basic solution to the heat diffusion equation and transient conduction analysis where we just changed the surface temperature conditions for slab. What we're going to do now we're going to look at a simplification technique that it takes us actually quite a long ways with many different types of analysis and that's called the lump capacitance method. Okay, so what the lump capacitance method does is it assumes that our object we're going to change the external condition on the object and that is by changing the convective heat transfer environment and what we're assuming is that there are no temperature gradients within the object and consequently we're assuming that the entire object is at one temperature and the entire object cools at whatever rate if we're doing a cooling or it could be heating in the case where we put it into a warmer environment. So let's begin with a little schematic and then we're all going to come up with an equation that enables us to characterize what is going on with these approximations. Okay, so what we have is we have some chunk of mass using a technical term. Here is our mass and at time less than zero the temperature of the mass is at temperature Ti and then what we do is we take that mass and we drop it into a liquid or it could be any other change in convective environment here I'm showing it as being going into a liquid but essentially what we're doing is we're changing the external convective heat transfer parameter or boundary condition on the object and with that assuming that this liquid is at a lower temperature than the initial temperature of the object so let's assume that we know that this object with time is going to cool when we put it into this environment and that's what we're interested in finding out. So let's take a look at how we can come up with an equation that describes what is going on and what I'm going to do I'm going to draw a little picture of our object over here so this is our chunk of mass we'll bring it over here and what I'm also going to do is I am going to say that the energy inside of that object will say it's energy storage but it could be changing with time and I'm going to put a control surface around that entire object and remember we've used control surfaces before and it enables us to quantify any energy transfer going across the surface the control surface itself and that will be the basis with which we'll use to come up with our equation and when we look at this object where is energy transfer going to be taking place well it's going to be taking place across that boundary and it is going to be leaving the convective heat transfer so we can then write that e out and I'll put a time rate there is going to be equal to the convective heat transfer value for that particular object so with that what we can do is we can express an energy balance across the control surface and the energy balance is going to simply be the energy leaving is equal to the change of energy within our chunk of mass and we know the energy leaving we said it's leaving via convective heat transfer so this has a minus sign so that's going to be minus ha and it's going to be the temperature of our chunk of material minus the ambient temperature of the liquid which is t infinity and that is going to be equal to the change in energy within the object itself and for that we basically use mc delta t per unit time so that is going to be the density times the volume that gives us the mass times the specific heat capacity times the change in temperature per unit time and so we get that there as being a differential equation now quite often what we do in heat transfer we want to convert these differential equations into homogeneous ordinary differential equations and in order to do that we make a substitution and we will introduce theta we saw this technique with fins we went through and introduced a theta value we quite often do that in heat transfer taking the derivative of that d theta by dt the time derivative okay so we get that now that's fine that's fine but what about this dt infinity by dt well what that is is we have to ask ourselves is the temperature of the liquid or whatever medium that we're putting it into changing with time and for lump capacitance we always assume that it's not we assume that it's much much larger and capable of storing energy or taking energy without changing in temperature and consequently we neglect that term there and we say that that's zero because the ambient or free stream temperature is not changing and so with that what we can do we can come back to our equation here making the substitution for theta let's rewrite our energy balance so we get that there now as here that is the surface area of our object the wetted surface area so what i'm going to do now i'm just going to rearrange some of the terms in here and we're going to put this into a form of an equation that we can integrate okay so we get that equation there just by rearranging and now what i'm going to do i'm going to integrate this equation and i'm going to apply the limits of integration what i'm going to integrate between so i'm integrating in time from zero to t oops sorry i forgot a minus there there should be a minus on that side i'm integrating from zero to t and in that time at time zero what we will say is that theta is theta i and then we're going to some value of theta that we're interested in so theta i would be with the initial temperature of the material theta i would be ti minus t infinity so that's what theta i is and what we can do we can integrate that d theta over theta is going to be natural logarithm so let's go ahead and do that and then the right hand side is just minus t and we can now introduce the limits for a natural logarithm and we end up with the following so we obtain that and now i'm going to take an exponential of both sides to get rid of the natural logarithm and i am going to reintroduce the substitution that we made for theta and so when i do that so we obtain that equation there and this is our solution this is an equation that tells us the temperature in an object as a function of time under the lump capacitance technique and so the temperature that we're usually after is right here and typically we will know everything else in this equation now one thing that we do now we introduce a thing called the thermal time constant so i will rewrite the equation in terms of the thermal time constant and i'll call that tau and so looking back at our solution it's basically just one over this term here is the thermal time constant and with that we can again rewrite our temperature and so it just makes it a little more compact quite often people will write it in terms of a thermal time constant but it tells you how quickly a system will respond to a change in in the in-pick condition in this case the convective heat transfer environment and what we can do we can plot that as a function of time and so if we have time and then on the vertical we put t minus t infinity divided by ti minus t infinity and we'll start at one because if you exponential of zero that is going to give us one so this here is zero and then what we can do we can do one time constant two time constants and just quickly plotting this if you put exponential of minus one and you take that you get point three six eight so we're going to be around here and then for two tau that would be about point one three five so you can see that fairly quickly the temperature in this system is going to drop and so it kind of look like that and then asymptotically it's going to go to zero as time goes to infinity so that would be what the response or the temperature would look like if you were to recast it in terms of the time constant instead of putting it for time so that is the lump capacitance technique and what we're going to do in the next segment is we're going to apply the lump capacitance technique to solve the problem and so it's kind of a handy quick technique there are restrictions and we will be looking at those in the next two segments we'll be talking about the restrictions for the lump capacitance technique but it enables us to get temperature in an object provided we can assume that the temperature within the object itself spatially is not changing that is we assume that it's a constant or lump temperature for the entire object so that is the lump capacitance method