CHAPTER 01.01: INTRODUCTION TO NUMERICAL METHODS On Solving an Engineering Problem - Part 2 of 2
The reason why we say that's not really true is because what happens is that, we use the formula for calculating the contraction. We use this particular formula, and this formula has a major assumption in it which is that alpha is a constant function of temperature or alpha does not change between the temperature of dry ice and alcohol mixture and the room temperature. But when we looked at the handbook and looked at some numbers which were given to us for alpha as a function of temperature we found out that alpha does change with temperature. So, your alpha does change with temperature. In fact, as you decrease temperature, alpha decreases. So that means that at lower temperatures the amount of contraction which will take place will be lower per unit of temperature change as opposed to at high values. That means that we may have overestimated the value delta D by using this particular formula here. So how do we go about finding out this contraction if we overestimated? Let's go and find out how much did we overestimated by. So if you look at this particular case here, now, the correct model, what the correct model will do is it will account for the thermal expansion coefficient varying. So, the correct model to use, or the correct equation to use to find out what the change in the diameter is as follows. Because alpha is no longer, cannot be assumed to be a constant function of temperature, where Ta is the room temperature and Tc is the temperature of the dry ice and alcohol mixture.
So let's go and see how can we now understand that hey, we simply have to take the area under the curve of alpha with respect to temperature as opposed to just multiplying the room temperature alpha by delta T to be able to calculate what delta D is. So what I would like you to do is, I would like you to roughly estimate the contraction. And one of the ways to roughly estimate the contraction is that, this is about minus 108 right here, this is 80 degrees Fahrenheit here, and what I will do is, I will just take, draw a straight line from there and just find out roughly how much this area under the curve is. So this area under the curve, the negative of this value will give me this value here. The reason why I say negative of that value is because I am integrating from 80 to minus 108. Since I am going from here, since the integral path is going from right to left, the amount of area which I have about the curve needs to be multiplied by a negative number to be able to calculate this number. So I want you to go ahead and do that, so that will give you the estimate of this, that will give you the estimate of that integral and then you can multiply by the diameter which is 12.363 inches and figure out what kind of value for delta D you will get. There is a, can you find a better estimate? Some people might say hey, I can get a better estimate. Again, what would like you to do as homework, is that say that hey, let's suppose I am able to find out what, let me do a linear interpolation here. I will be able to find out what the value of alpha at minus 108 degrees Fahrenheit is and I will take this trapezoid and I will take this trapezoid here and I will take this trapezoid here and I will take this trapezoid here because that is at 80 degrees Fahrenheit. And what I will do is, I will take, find the area of this trapezoid and I will find the area of this third trapezoid and the area of this fourth trapezoid and I can add all those areas of those four trapezoids to be able to calculate what the area under the curve is. Again, I will have to take the negative of that because I am going from right to left and so far as the limits of integration are concerned. I multiply by the diameter, will give will give me what the change in delta D is. So, this is again part of your homework and you will go ahead and find out what answer do you get.
Now, the approach which we took was that in order to find out the contraction accurately is that we took the data which was given to us and we found a regression line by using Matlab and we will talk about regression in the course itself. We took, found out the value of alpha turns out to be a second order, regresses to a second order polynomial. That is the approximation which we get for alpha as a function of temperature by regressing the data which was given to us to a second order polynomial. So, what that means is that we can take the second order polynomial, substitute it in here, and everybody knows how to integrate a second order polynomial from 80 to minus 108 and that will give us the value delta D. And that turns out to be minus 0.0137. So what we are finding out is that the amount of contraction in the diameter of the Trunnion has been underestimated when we use the room temperature alpha instead of understanding that alpha changes as a function of temperature. So, what is the solution to the problem? Because we are not able to solve the problem. We have basically said that hey, we underestimated the, we underestimated the contraction. So what we can do now is that we can immerse the Trunnion in liquid nitrogen. The reason why we say hey, this is a better alternative than using dry ice and alcohol mixture is because the boiling temperature of liquid nitrogen is minus 321 degrees Fahrenheit as opposed to dry ice and alcohol mixture which is at a temperature of minus 108. So that will give us more contraction in the diameter. So again I will ask you to do this as a homework, to figure out how do we get delta D. The delta D, the contraction that we get is minus 0.0244 inches which is a lot more than the 0.015 inches which is required for our specifications. Let me review the four steps which we talked about in solving an engineering problem. So we had a problem, so what is the problem? That the Trunnion got stuck in the Hub. We didn't want that problem to take place again so we developed a new model. We know that hey, even when we develop a mathematical model we have to understand that we are developing a good accurate model to be able to do so. So we already know that using delta D is equal to D times alpha times delta T is not a good idea because alpha is changing as a function of temperature. So we are going to use a more accurate model which is as follows: we solve the problem. I assigned you some problems by using trapezoidal rule, by drawing those trapezoids which I showed you or using regression integration which is something which I showed you how to do. But I am asking you to do this as your homework. And then we have implemented the solution.
So the implementation of the solution was not saying that hey, no we cannot do anything about it. The implementation of the solution was that hey, we need to find a different cooling medium. And one of the solutions can be to use liquid nitrogen. So if you use liquid nitrogen it is enough to contract enough to be able to use for the assembly. And that's the end of this segment. |