Hello, and welcome to lecture number 33 of this lecture series on turbo machinery aerodynamics. As promised in the last lecture, we shall be having a tutorial session in today's lecture. But before I start the tutorial, let me quickly recap what we have been discussing in the last two lectures. As you know that last two lectures have been devoted exclusively for centrifugal compressors. So, during the first lecture on this series that was on lecture number 31, where we introduced the aspect of centrifugal compressors, we discussed in detail the thermodynamics of centrifugal compressors. And we also discussed in some detail about why is it that centrifugal compressors have certain amounts of benefits in some cases and as well as the fact that there are lot of disadvantages associated with centrifugal compressors and that is primarily the reason why these compressors are not used in large sized engines. So, the application of centrifugal compressors are primarily at this moment limited to smaller sized or smaller thrust class engines and because axial compressors have inherent advantages over centrifugal compressors, which makes them more suitable for larger sized engines. But centrifugal compressors continue to be used in smaller engines and of course, they were used popularly in the early days of development of the jet engines when axial compression system was not that developed in those days. And then we also discussed about the different components, which constitute a centrifugal compressor like the inlet part of the centrifugal compressor, the inducer. Induser is one which guides the flow into the impeller to ensure that there is smooth entry of the flow into the impeller. Then the impeller itself, which is the rotor of the centrifugal compressor. Induser is often attached to the impeller and it forms the initial part of the impeller itself. Of course, in some of the older generation engines and compressors, inducer was sometimes kept as a separate component and not necessarily a part of the impeller itself. Impeller flow gets discharged into a vaneless space, which is where the diffusion continues from the impeller, it then continues in the vaneless space. We have seen the physics behind that and then the flow proceeds into the diffuser, which could be either vane type diffuser or it could be pipe diffuser, channel diffuser and so on. From the diffuser, the flow is collected through what are known as collectors or volute and then it is discharged from the compressor. So, these are different components, which constitute a centrifugal compressor. We have seen the flow as it passes through these different components, how we can calculate different parameters associated with the flow. So, these were some of the things we had discussed in the first lecture, which was on during lecture number 31. In the previous lecture, we continued our discussion on this on little more advanced concepts like the Coriolis acceleration and what its significance is in reference to centrifugal compressors. We have seen that Coriolis acceleration basically leads to a tangential variation in the relative velocity and at the exit of the impeller, this would mean that there is what is known as a slip. Slip is basically referring to a difference between the tangential component of the absolute velocity to the tip blade speed that is C w 2 by u 2 and so slip factor basically affects the performance in terms of pressure rise of the centrifugal compressor. We have seen that slip factor is a strong function of the number of blades. Therefore, there are empirical correlations, which from which one can calculate slip factor for different types of impellers and subsequently, we discussed in detail about the performance characteristics of centrifugal compressor. We have seen that it is very similar to that of axial compressors and we spend some time discussing about the choking aspect of centrifugal compressor and also the fact that choking is different in different components that is the way choking is calculated in the impeller or the that is the rotating components or the stationary component like the inlet or the diffuser is quite different and also the fact that one can continue to operate the engine at a higher mass flow than the choking mass flow, if we can increase the speed of the impeller and also the fact that this does not lead to choking of the other components that is as long as all the other components are not choked for a given operating condition, if the compressor is operating under choked condition, if we increase the rotational speed we can actually operate at a slightly higher mass flow rates at least theoretically. These were the different aspects that we had discussed in the last two lectures and therefore, it is about time that we have a tutorial session and I think I also mentioned why we are not emphasizing too much on the centrifugal or the radial flow machines like centrifugal compressors or you will see shortly about radial turbines is the fact that these components or these types of turbo machines though have been used extensively in the past in the current day scenario their application is very much limited and most of the modern day jet engines that we are aware of the larger sized ones definitely use axial flow turbo machines like the compressors and turbines. So, today's lecture we will devote towards centrifugal compressors on basically trying to solve some problems on centrifugal compressors. Let us look at basically have a tutorial session on centrifugal compressors. So, we will have a tutorial on centrifugal compressors. Let us take a look at the first problem that we have today. So, the first problem statement is the following at the inlet of a centrifugal compressor I the relative Mach number is to be limited to 0.97 the hub to tip radius ratio of the inducer is 0.4 the I tip diameter is 20 centimeter and if the inlet velocity is axial determine part A the maximum mass flow rate for a rotational speed of 29 160 rpm part B the blade angle at the inducer tip for this mass flow the inlet conditions can be taken as 101.3 kilo Pascal's and 288 Kelvin. So, let us read this question little more carefully what is basically the problem statement referring to it tells us that there is a relative Mach number at the inlet I and that is limited to 0.97 that is we have to keep the Mach number limited to 0.97 and not let it attain supersonic speech the half to tip ratio is given as 0.4 the I tip diameter is given as 20 centimeter then we have the blade rotational speed we need to calculate maximum mass flow and the blade angle. So, let us take a look at the schematic of this particular problem we have this half to tip ratio that is been given to us the radius ratio is 0.4 which means that R H by R T is 0.4 the inlet velocity is given as axial. So, C 1 is axial and then the blade angle beta 1 is one of the parameters we need to calculate since u 1 is also specified because we know the tip diameter and the rotational speed. So, from there we can calculate the blade angle beta 1. So, this basically is a problem which refers to a typical centrifugal compressor problem which has the flow entering the inducer axially in this case the flow is indeed axial and that is one of the parameters which has been given rotational speed is given some of the geometrical dimensions with reference to the inducer is also given that the tip diameter the half to tip ratio etcetera is given to us. And one of the important statements in the problem is that we need to ensure that the Mach number is limited to 0.97 at the inlet actually start solving the problem from there onwards. So, we know that the Mach number is to be limited to that and then from there we can calculate let us say the static temperature because inlet stagnation temperature is given the inlet velocity is basically can be calculated. So, if you look at the rotational speed we have been given that the rotational speed is 29 160 rpm diameter is 0.2 20 centimeters. And so, we have pi d n by 60 this is 305.36 meters per second. Now, from the velocity triangle we can see that the relative Mach number is basically relative velocity by square root of gamma r t 1 and at the inlet v 1 is equal to square root of c 1 square plus u 1 square this divided by square root of gamma r t 1. And so, we also know that t 1 is equal to t 0 1 minus c 1 square by 2 c p and t 0 1 is given as 288 c 1 of course, is not known c p we will assume for air as 1005 joules per kilogram Kelvin. And therefore, 2 c p is 2010. So, here we have 2 equations one is for Mach number and the other is for static temperature. So, let us substitute this in this equation relative Mach number becomes square root of c 1 square plus u 1 square divided by square root of gamma r into the temperature 288 minus c 1 square by 2010 right hand side is involves an unknown that is c 1 and left hand side is already given that we have to limit the Mach number to 0.97. So, if you substitute all the values we have 0.97 square is equal to c 1 square plus 305.63 square divided by gamma r into 288 that is 115718.4 minus 0.2 c 1 square. So, if you simplify this it is a simple quadratic equation which we can simplify and then we can get c 1 is 114.62 meters per second. So, if you refer to the velocity triangle we have basically determined c 1 and since u 1 is known you can now calculate beta 1 because it is basically the time of u 1 and c 1 of course, the first part of the question is also to find the mass flow rate. So, let us calculate the mass flow rate for calculating mass flow rate we need the density axial velocity is known we also need the area. Now, the static temperature you can calculate here because the stagnation temperature is given c 1 we have just now calculated and therefore, static temperature would be 288 minus c 1 square by 2 c p c 1 square we have calculated and therefore, it becomes 281.46. And then since no efficiency has actually been specified we can use isentropic relations here to calculate the corresponding static pressure because stagnation pressure has been specified stagnation temperature and static temperature are known. So, if you substitute these values here we can calculate the static pressure p 1 which is 93.48 kilo Pascal's. Now, once we calculate the pressure and temperature we can calculate the density which is p 1 by R T 1 and that is 1.157 kilo grams per meter cube. So, this is one of the parameters which we need for calculating mass flow rate the other parameter is the area of passage of the mass flow rate and that is the annulus area at the inlet for which we have been given the tip diameter and the hub to tip ratio. So, the annulus area would be pi d square by 4 into 1 minus R H by R T. So, that is 0.0264 meter square. So, this is the annulus area through which the mass flow rate passes and we have calculated density. Now, the third parameter for calculating mass flow rate is the velocity and for mass flow rate we need to know the axial velocity, but at the inlet we know that the absolute velocity itself is axial and therefore, C A 1 should be equal to C 1. So, mass flow rate is quite straight forward now we just multiply density annulus area at inlet and the axial velocity. So, rho 1 A 1 and C 1 the product of these 3 would give us the mass flow rate and so we multiply the density area and absolute velocity we get the mass flow rate. Then the second parameter is the blade angle. So, how does one calculate the blade angle? Blade angle at the inlet is very straight forward again because we have already calculated C 1 and we know the blade speed u. So, tan inverse of C by u C 1 by u 1 would give us the angle at the inlet that is beta 1. So, if you substitute those values here we get mass flow rate as rho 1 A 1 C 1 which is 1.157 into 0.0264 which is the annulus area C 1 is 114.62 meter per second. So, the product of all the 3 will give us the mass flow rate which comes out to be 3.5 kilograms per second blade angle at the inlet is at the tip is tan beta 1 which is C 1 by u 1. Therefore, beta 1 is tan inverse C 1 by u 1 which comes out to be 20.57 degrees. So, these are two parameters that we have calculated the mass flow rate which was the first part of the question. Second part was for the same mass flow what is the blade angle and so we have also calculated the blade angle for this question. I think I have mentioned this several times in the past that the key to solving problems to do with turbo machines the fundamental problems to do with turbo machines is to get the velocity triangles. So, even for a centrifugal compressor as you have seen velocity triangle is the starting point for any of this analysis. So, you start solving the problem once you get the velocity triangles. So, if the velocity triangles can be set right then solving the problem is quite easy because you know what to calculate and how to calculate them. So, I would urge you to keep this in mind whenever you are solving a problem irrespective of whether it is axial compressors axial turbines centrifugal or radial turbines. Whatever be the problem that you are solving it is very important that you understand the significance of velocity triangles and that is why you need to understand the physics very well to be able to construct the velocity triangle and that would basically be the starting point for solving any of these problems. So, let us move on to the next problem now and let us see what this problem statement is. Second problem statement is that a centrifugal compressor has a pressure ratio of 4 is to 1 with an isentropic efficiency of 80 percent when running at 15000 rpm and inducing air at 293 Kelvin curved vanes at the inlet give the air a pre whirl of 25 degrees to the axial direction at all radii. The tip diameter of the eye of the impeller is 250 millimeters the absolute velocity at inlet is 150 meters per second and the impeller diameter is 600 millimeters calculate the slip factor. So, this is in fact a follow up question for this will also will be given a little later we will also solve that problem which is quite similar to this, but with a small twist. So, in this case the question states that we have a centrifugal compressor which is developing a pressure ratio of 4 is to 1 with a certain efficiency and rotational speed what is to be noted here is that the air is no longer entering axially. Unlike the previous question where it was specifically mentioned that the inlet air is axial here it is not axial it is coming with a pre whirl that is a pre whirl is like guide vane ahead of the inducer where the guide vane set the flow at a certain angle in this case the angle has been set at 25 degrees. And then we have been given some dimensions of the impeller and some speeds. So, based on this we need to calculate the slip factor. So, slip factor as we have learnt in the previous lecture is basically the ratio of the tangential absolute velocity at the impeller exit divided by the corresponding blade speed at the same location. So, it is C w 2 by u 2 which is the slip factor we are required to find this. Now, since our rotational speed is given and the diameter of the impeller is given finding the blade speed at the tip is very easy because u 2 is pi d 2 n by 60 and d n both are given to us. And so you can calculate the impeller tip diameter tip speed, but how do you calculate C w 2? This of course, will require us to solve the velocity triangle first at the inlet and subsequently at the exit as well to be able to solve this problem. So, let us look at the velocity triangle at the inlet of the inducer. So, these are the inducer veins which are shown and you can see that the flow is not entering the flow in inducer in an axial direction it is in fact entering at an angle of 25 degrees. That means that C 1 is not axial C 1 itself has a tangential components C w 1 then the inlet relative velocity is v 1 and the blade speed at the inlet is u 1. So, this is which means that there would be a set of veins which will set this flow angle of 25 degrees to ensure that the absolute velocity enters the inducer in this direction and that is called a pre whirl. You can see this word mentioned here it is called pre whirling. Pre whirl means that there is a whirl component which is basically the tangential component as of absolute velocity and that is why it is called a pre whirl. In the absence of pre whirl the flow would have entered axially, but that may not really be an ideal condition for the inducer because the relative velocity might actually enter the inducer at a different angle of course, this has been designed for a pre whirl. So, let us begin to solve this problem we have the pressure ratio given to us and therefore, we can calculate the exit stagnation temperature exit stagnation temperature is equal to T 0 1 times the pressure ratio raise to gamma minus 1 by gamma and therefore, we get 293 into 4 raise to 1.4 minus 1 by 1.4 that is 435.56 Kelvin. So, the isentropic temperature rise would be 435.56 minus T 0 1 that is 293 we get 142.56 Kelvin, but the actual temperature rise is isentropic temperature rise divided by the efficiency which in this case is specified as 80 percent and therefore, we get delta T naught is equal to 178.2 Kelvin which is 142.56 divided by 0.8. Therefore, from the temperature rise we can calculate the work done per unit mass and work done is can be calculated in two ways as you have seen in the past. One is by using the temperature rise and the other method is by using velocity triangles that is you delta u C w will basically give us the work done per unit mass. And so here we know since we know the temperature rise we can calculate the work done per unit mass which is basically the enthalpy rise in the compressor and that is C p times delta T naught. Delta T naught we have calculated as 178 this multiplied by C p will give us the work done per unit mass. So, work done here would be C p which is 1.005 into delta T naught delta T naught is 178.2 therefore, we get 179 kilo joules per kilogram. Now at the exit we can now calculate some parameters at the exit because our aim is to calculate the slip factor at the exit and but of course, before that we need to calculate C w 1 because that is required for calculating the exit parameters as well. Now the inlet of the i we have u 1 is pi d n by 60 that is pi into the tip of the i of the impeller is 0.25 meters into 15000 divided by 60 and that comes out to be 196.25 meters per second. C w 1 is C 1 times sin 25 C 1 is given to us and therefore, and also the angle is given alpha C 1 sin alpha 1 therefore, that comes out to be 63.4 meters per second. Now peripheral velocity at the tip of the impeller u 2 would be pi into capital D which is the tip diameter of the impeller that is given as 600 millimeters multiplied by n by 60. So, pi into 0.6 into 15000 divided by 60 this is 471.2 meters per second. So, here we have calculated the conditions at the inlet which refers to the C w 1 and why we are calculating C w 1 is because we have just now calculated the work done per unit mass. And work done per unit mass is also equal to u 1 u 2 C w 2 minus u 1 C w 1 and from there u 1 and u 2 are known C w 1 we have just now calculated and therefore, we can calculate C w 2. Once you calculate C w 2 ratio of C w 2 to the blade speed at the tip that is u 2 will give us the slip factor. So, let us now calculate C w 2 which is the world velocity at the tip of the impeller and since the power work done per unit mass is also equal to u 2 C w 2 minus u 1 C w 1 and this is equal to work done we have just calculated earlier 179 kilo joules per kilogram. This is equal to u 2 times C w 2 that is 471.24 into C w 2 minus u 1 C w 1 that is 196.35 into 63.4. So, C w 2 we can calculate as 406.27 and therefore, we calculate slip factor as ratio of C w 2 to u 2 and therefore, 406.27 divided by 471.24 that is 0.862. So, here we get a slip factor of 0.86 what are the implications of this? Well the implications are depends upon the kind of application we have, but the basic implication of having a slip factor much lower than 1 is that it leads to lower and lower pressure ratios that is higher the slip factor lower is the pressure ratio for which it is actually been designed for. This is because the tangential velocity at the exit that is basically the solid velocity at the exit is directly related to the pressure ratio. You can actually derive an expression which relates the pressure rise P 0 2 by P 0 1 in terms of C w and u 1. So, therefore, there is a direct correlation between that. So, lower the slip factor which means lower is C w 2 in comparison to u 2 and therefore, that will affect the pressure rise achieved in a centrifugal compressor and that is why modern day designers would like to maximize or increase the value of slip factor to keep it as close as possible to 1. We have seen the strong dependence of slip factor on the number of blades and that is one of the key optimization challenges because one can keep increasing number of blades, but then the problem with that would be the fact that increasing number of blades also leads to an increase in these skin friction losses and therefore, that is going to affect the efficiency in some way. The designer does not want to have a poor efficiency with a higher pressure ratio that does not make sense. So, one needs to have a choice of will basically a mix of high efficiency as well as higher pressure ratio and that requires a very intelligent way of trying to optimize this case where we have one on one hand one option of increasing number of blades with the risk of lower efficiency because of losses. Other option is to reduce number of blades and have higher efficiency possibly higher efficiency because of lower frictional losses, but one may end up with a lower pressure ratio. So, there is a trade off required to attain an optimum condition. So, let us now again look at another problem, third problem also involves slip. Let us take a look at what the problem statement is. The problem number 3 states that air at a stagnation temperature of 22 degree Celsius enters the impeller of a centrifugal compressor in the axial direction. The rotor which has 17 radial vanes rotates at 15,000 rpm. The stagnation pressure ratio between the diffuser outlet and the impeller inlet is 4.2 and the total to total efficiency is 0.83 percent or 0.83. Determine the impeller tip radius assuming that the air density at the impeller outlet is 2 kgs per meter cube and the axial width at the entrance to the diffuser is 11 millimeters. You also need to determine the absolute Mach number at this point. We assume that the slip factor is 1 minus 2 by n where n is the number of blades. So, we have been permitted to use the standard slip factor which states that the slip factor is equal to 1 minus 2 by n n being the number of vanes. So, in this question we have been given the number of blades here 17 radial vanes rotational speed 15,000 rpm and the stagnation pressure ratio 4.2 total to total efficiency 83 percent and then the density at the impeller outlet and the axial width at the entrance of the diffuser. So, these are the parameters which are given to us based on which we need to calculate two parameters. One is the impeller tip radius and the second is we also need to calculate the Mach number at the exit of the impeller. So, we need to of course, calculate the absolute Mach number. So, this is a question which is very similar to one of the questions we have solved earlier. So, I will skip the velocity triangle part leaving that to you to figure out the velocity triangle and try and construct the velocity triangle for that case. So, this question involves requires us to calculate two parameters one of course, the hint given to us is that we can calculate slip factor using the Steinitz formula we have we have been given the number of blades. So, we can actually calculate the slip factor from there and then we know the pressure ratio and total to total efficiency and. So, we can actually calculate the power required from what data has been given to us. The other key information is that the flow enters the impeller in the axial direction which means that there is no tangential component to the absolute velocity at the inlet. So, C w 1 should be 0 and so you can calculate work done based on just u 2 times C w 2. So, the specific work required is u 2 C w 2 minus u 1 C w 1 C w 1 is 0 and therefore, work is u 2 C w 2 which is also equal to sigma times u 2 square because sigma is C w 2 by u 2 and. So, we can express work done in terms of sigma which is the slip factor and the blade speed at the tip of the impeller u 2. So, sigma u 2 square would be the work done specific work done. So, we can now express efficiency in terms of well u 2 in terms of efficiency and pressure ratio. This comes from the total to total efficiency we have already defined that earlier on in the context of a turbine. So, similar aspect we can also apply for this case. So, when we use the efficiency definition we have the outlet temperature T 0 3 minus T 0 1 T 0 3 being the isentropic temperature minus T 0 1 divided by T 0 3 minus T 0 1 and then the pressure ratio is given to us. So, the numerator gets converted or can be expressed in terms of the pressure ratio. Pressure ratio has been given to us as 4.2 and the denominator which is T 3 minus T 0 1 is basically the work done divided by specific heats. So, you can express that in terms of w times w divided by C p w is sigma into u 2 square and therefore, we can express u 2 in terms of these parameters which are known to us the inlet stagnation temperature sigma the temperature and pressure ratio as well as the efficiency. So, if you express u 2 in terms of efficiency and pressure ratio we have u 2 square is C p T 0 1 multiplied by pi c raise to gamma minus 1 by gamma minus 1 divided by sigma into efficiency. Efficiency is given as 83 percent and sigma is given can be calculated based on the number of blades. So, sigma is 1 minus 2 by n, n is 17 we have 17 number of vanes of blades. So, sigma the slip factor is 0.8824. So, we know the temperature you know pressure ratio and the efficiency if you substitute that we get the blade speed as 452 meters per second. And then omega because the speed is given to us as 15000 rpm based on that we can now calculate the tip radius because tip radius is u 2 or the tip speed is basically equal to omega into r. Therefore, r t is equal to u 2 by omega and omega here is the rotational speed in radians per second 15000 into pi by 60 we get 1570 radians per second. So, if you substitute for those two 452 divided by 1570 we get the tip radius as 0.288 meters. So, this is the first part of the question where we are required to find tip diameter of the impeller which in this case comes out to be 288 millimeters or 0.288 meters. The second part of the question is to find the Mach number well the absolute Mach number at the impeller exit which means that we have to take the ratio of C 2 to the speed of sound at that location that is A 2. So, C 2 by A 2 will give us the Mach number at station 2 which is impeller exit. And so we need to now find out the absolute Mach number we also need to know the temperature static temperature at the exit of the impeller to be able to calculate the Mach number. Now, at the exit of the impeller we have C 2 which is root of C w 2 square plus C r 2 square where C r is the radial velocity the absolute component of radial velocity. And how do we calculate C r 2? C r 2 would basically be coming from the mass flow rate mass flow rate divided by the density times the annulus area. Now, in this case we have mass flow rate which has been given to us as 2 kilograms per second density is also given as 2 kg per meter cube tip radius we have just now calculated 288 millimeters or 0.288 meters. And the axial width is given to us in the question as 11 millimeters. So, this is 0.011. So, C r 2 which is the radial velocity at the exit of the impeller is mass flow rate divided by density times the annulus area 2 pi into tip radius times the axial width which gives us the annular area this comes out to be 50.3 meters per second. Similarly, C w 2 we can calculate because we know u 2 we also know the slip factor sigma product of those 2 we would give us the tangential velocity at the exit C w 2. So, this is 400 meters per second. So, C 2 is equal to square root of 50.3 square plus 400 square that is 402.5 meters per second. So, for calculating Mach number we now have the absolute velocity we now need to calculate the static temperature as well. So, for calculating the static temperature we will make use of the rothalpy we have already discussed about that earlier on. So, we will make use of that concept here the conservation of rothalpy in the impeller and based on that we will calculate the static temperature. So, we know that h 0 2 which is stagnation temperature at the exit of the impeller would be equal to inlet stagnation temperature plus the work done that is w c. So, h 0 1 plus w c will be h 0 2 and therefore, h 0 1 plus sigma u 2 square because w c we know is in this case is sigma u 2 square and therefore, h 2 would be equal to h 0 1 plus sigma u 2 square minus half c 2 square because h 2 plus half c 2 square is h 0 2. This we will convert in terms of temperatures since h 2 is equal to h 0 1 plus sigma u 2 square minus half c 2 square correspondingly in terms of temperature we get t 2 is equal to t 0 1 plus sigma u 2 square minus half c 2 square divided by c p. So, we substitute all these values and we get temperature static temperature at the impeller exit as 394.5 Kelvin. Therefore, the Mach number at the exit of the impeller is 402 which is the absolute velocity divided by square root of 1.4 into R 287 into 394.5 Mach number comes out to be 1.01. So, we can see that the absolute Mach number is just about sonic it has just crossed the sonic Mach number and in fact, in the relative frame of reference the Mach number can in fact be higher than what you have calculated for the absolute case here. So, the third problem we just now solved concerned about was basically about calculating firstly the Mach number we have seen how to calculate the Mach number as well as how do you calculate the based on the parameters like the slip factor in this case to be approximated as 1 minus 2 by n. We can also calculate the mass flow rate as we have done in this question. Now, let us take up one more problem which is in some sense identical to what we have solved in the second question and partly also identical to what we have just now solved for the third question. So, we will require understanding of how those two problems were solved to be able to solve this question. So, we will be calculating the slip factor as well as the Mach number in this question that we are going to solve. Let us take a look at the problem statement for this question. A centrifugal compressor with a backward leaning blades develops a pressure ratio of 5 is to 1 with an isentropic efficiency of 83 percent. The compressor runs at 15000 rpm, induces are provided in the inlet of the compressor. So, that air enters at an absolute velocity of 120 meters per second. The inlet stagnation temperature is 250 Kelvin and the inlet air is given a pre whirl of 22 degrees to the axial direction at all radii. The mean diameter of the i of the impeller is 250 millimeters and the impeller tip diameter is 600 millimeters. Determine the slip factor and the relative Mach number at the impeller tip. So, you can immediately see that it has components of both the second question as well as third question. Second question we actually calculated the slip factor and in third question we calculated the absolute Mach number. Of course, in this case we are required to calculate the relative Mach number. Let us look at the velocity triangles first both at the inlet as well as the exit. This is the velocity triangle that you should get at the inlet and you have a schematic of the inducer and also a fixed inlet guide vane which gives a pre whirl to the inlet flow which is entering the inducer. So, these guide vanes put which are set at an angle of alpha 1 gives a pre whirl which is given in this case as 22 degrees causing the absolute velocity not to be axial. And therefore, the absolute velocity C 1 is not axial unlike some of the questions we have solved earlier on. And that causes the velocity triangle at the inlet to look like what is shown here. This is C 1 the relative velocity v 1 and the blade speed u 1. At the exit on the other hand these are backward leaning blades and so the velocity triangle for a backward leaning blade should look like this. Since, the blades are leaning like this the relative velocity leaves the blades in this direction with an angle of beta 2. And this is the absolute velocity C 2 and the blade speed u 2 at the tip of the impeller. The axial velocity component is shown here as C A and the rotational speed is omega which is given in this case as 15,000 r p. Let us first try to solve the inducer part of the question and then we will move towards solving the second part. Now, the inlet temperature is given as 300 Kelvin and therefore, based on the pressure ratio and isentropic relation we can calculate the exit stagnation temperature isentropic. And that is T 0 2 S which is equal to T 0 1 into pi c raise to gamma minus 1 by gamma. And this is 250 multiplied by 5 raise to 0.4 by 1.4 that is gamma minus 1 is 0.4 divided by 1.4 it is basically 395.95 Kelvin. Therefore, the stagnation temperature isentropic is 395.95 minus 300 that is 95.95 Kelvin. But the actual temperature rise is this divided by the efficiency. In this case efficiency is given as 0.83 and therefore, T 0 S actual basically would be equal to this divided by efficiency that is 95.95 divided by 0.83 that is 115.6 Kelvin. So, the specific work required would be C p times delta T 0 that is 1005 into 115.6 this is 116.186 kilo joules per kilogram. Now, since it is given that C 1 is 150 we can calculate C w 1 which is the world component of the absolute velocity that is C 1 times sin alpha 1 alpha 1 is 22. Let me go back to the velocity triangle here C 1 times sin alpha 1 would give us C w 1 which is this component. So, that comes out to be 56.2. Now, here what we are going to do is since we know the specific work done from the temperature rise we also know that specific work done is equal to the product of u 2 times C w 2 minus u 1 times C w 1. C w 1 we have calculated we can now calculate u 1 and u 2. Therefore, we get C w 2 and once we know C w 2 we can calculate the slip factor which is C w 2 by u 2. So, let us first calculate u 1 and u 2 u 1 is pi d mean of the i of the impeller into the rotational speed by 60 that is given as pi into 0.25 into 15000 by 60 this is 196.3 meters per second and u 2 is the blade speed at the tip of the impeller pi d t into n by 60 that is pi into 0.6 into 15000 by 60 471.24 meters per second. Now, specific work done as we have seen we calculated that already from the previous calculation specific work is C p delta t naught that is 116.186. Specific work is 116.186 into 10 raise to 3 is equal to u 2 which is 471.24 into C w 2 minus u 1 which is 196.3 into C w 156.2. So, from this we can calculate C w 2 that is 269.96 meters per second. Therefore, slip factor is the ratio of this C w 2 divided by u 2 that is 269.96 divided by 471.24 that is 0.573. You can see that the slip factor is a very low number here one would normally expect a relatively higher slip factor of the order of 0.7.8 or even higher than that. This is a very low slip factor and we have already discussed the disadvantages of having very low values of slip factor basically affecting the pressure rise of the centrifugal compressor for a given rotational speed. So, the next part of the question is to calculate the Mach number at the tip in the relative frame. So, the relative Mach number at the tip of the impeller for which we will refer to the velocity triangles once again. Let us look at just the exit velocity triangle. For a backward leaning blade we have seen that the velocity triangle would look like this. We have the absolute velocity C 2 and the relative velocity V 2 which is leaving the blades tangentially. So, this is how the exit velocity triangle would look like we need to calculate V 2 and also the temperature at the exit of the impeller to calculate the Mach number. So, from the velocity triangle we see that V 2 is equal to square root of C A square plus U 2 minus C w 2 the whole square. So, this C A at the inlet of course, we are assuming that the axial velocity does not really change as it passes through this impeller. This is equal to C A square that is C 1 cos alpha 1 the whole square minus well plus U 2 minus C w 2 the whole square. Since, all these parameters are known we substitute that we get the relative velocity 222.9 meters per second and then we need to also calculate the static temperature at the tip T 2 which is T 0 2 minus C 2 square by 2 C p. T 0 2 we can calculate because we know the efficiency and the pressure ratio. So, from there we calculate T 0 2 which is T 0 1 plus T 0 2 s minus T 0 1 by efficiency and that comes out to be 365.61 and we also need to calculate C 2 from the velocity triangle we see that C 2 is C 2 square is equal to the this component C w 2 square plus C A square. So, C w 2 we have already calculated that is 269.9 that square plus C A square which we can calculate from this C 1 cos alpha 1 that is 139.08 square. So, C 2 comes out to be 303.68 meters per second. Therefore, the static temperature is equal to 365.61 minus C 2 square by 2 C p therefore, T 2 comes out to be 319.73 Kelvin. Therefore, the relative Mach number is the relative velocity divided by square root of gamma R T 2 and this if you substitute we get a relative Mach number of 0.62. So, the relative Mach number at the impeller tip in this case is calculated as 0.62. So, this completes four problems that we have solved in today's class. We started off with a very simple problem which just involves solving the velocity triangle and calculating the velocities and also the angles involved. Second question was to calculate basically the slip. Third question involved calculating the Mach number in the absolute frame and the last question was a combination of the second and third calculating the slip factor as well as the Mach number at the tip. So, these were four questions that we have solved in today's lecture. I now have a few exercise problems which you can take up and solve based on our discussion today as well as what we have discussed during the lectures. Let us take a look at the first exercise problem. The first exercise problem states that the design mass flow rate of a centrifugal compressor is 7.5 kg per second with the inlet stagnation temperature of 300 Kelvin and pressure of 100 kilo Pascal. The compressor has straight radial blades at the outlet. The blade angle at the inducer inlet tip is 50 degrees and the inlet up to tip ratio is 0.5. The impeller is designed to have a relative Mach number of 0.9 at the inducer inlet tip. If the tip speed is 450 meters per second, determine part a the air density at the inducer inlet, part b inducer inlet diameter, part c the rotor rpm and part d the impeller outlet diameter. The answers to these four different parts of the questions are density should come out to be 0.988 kgs per meter cube, inducer inlet diameter is 0.258, rotor rpm is 17100 rpm and impeller outlet diameter is 0.502 meters. The second exercise question is a centrifugal compressor runs at 10000 rpm and delivers 600 meter cube per minute of air at a pressure ratio of 4 is to 1. Isentropic efficiency of the compressor is 0.82. The outlet radius of the impeller is twice the inner radius, axial velocity is 60 meters per second. If the ambient conditions are 1 bar and 293 Kelvin, determine part a the impeller diameter at inlet and outlet, the power input the and the impeller angles at the inlet. The answer to the questions are impeller diameter at the inlet should be 0.92 meters and outlet 0.461 and the power input is 2044 kilo watts, impeller and diffuser angles at the inlet are 13.9 degrees and 7.1 degrees. Third exercise problem is 30 kilograms of air per second is compressed in a centrifugal compressor at a rotational speed of 15000 rpm. Air enters the compressor axially and the compressor has a tip radius of 30 centimeters. Air leaves the tip with a relative velocity of 100 meters per second at an angle of 80 degrees. Assuming an inlet stagnation pressure and temperature of 1 bar and 300 Kelvin respectively, find part a the torque required to drive the compressor, the power required and the compressor delivery pressure. So, the torque in this case will be 4085 Newton meters, power required 6.417 megawatts and compressor delivery pressure is 6.531 bar. And the last problem is a centrifugal compressor has an impeller tip speed of 366 meters per second. Determine the absolute Mach number of the flow leaving the radial vanes of the impeller when the radial component of velocity at impeller exit is 30.5 meters per second and the slip factor is 0.9. Given that the flow area at the impeller exit is 0.1 meter square and the total to total efficiency is 90 percent determine the mass flow rate. So, in this case we have the absolute Mach number as 0.875 and mass flow rate as 5.61 kg per second. So, these are 4 exercise problems that you can solve based on what we have discussed in the last 3 lectures including today's. And I hope based on these discussions you will be able to solve these 4 exercise problems that we have for you today. And we will continue our discussion on some of these topics especially some of the radial flow machines. We will probably be taking up the radial flow turbines in the coming lectures. So, we will discuss more to do with the radial flow machines in some of the coming lectures in the next few lectures.


Okay, I think I have to sit down a bit. All right, so welcome back to BC 106, our second lecture today. We're going to spend a little bit of time looking at some of the tools that are available for us to do Bible study. The first tool that I want to share with us is it's called eSword, all right? Let me, it's in the document. So I'll just share that and then I'll switch back to my screen here. So you can download this. It's free from eSword.net and this tool has been around for a long time. So I've been using it from 2001, almost 20 plus years. Okay, so it's a very useful tool, right? Then there are other tools, which we will talk about, but I would encourage you. Now, there are other tools, so I'm not trying to promote only this tool. I'm just sharing this one first because I use it and I've used it for almost 20 plus years and it's very good, but there are other tools which we will mention, okay? So you use whatever you're comfortable with. This is something I've been very comfortable with. Also, it's much simpler than the other tools. The other tools are there, they're very elaborate. So it has a lot more information, other tools, but that makes it that much more difficult to study to use. This one is very simple, easy to use. So let me, if I sit down, they won't see my face, but that's okay, right? Cause you're gonna look at the screen, right? So let me just share my screen. I'm gonna share my screen so that I can just walk you through this tool. So I have this tool, let me just, okay, I have this tool set up right here, so I launch it. It's free on Windows, right? So for those, but I think they have a paid version for Mac. So, all right, let me, I'll switch to my laptop camera. The online students, can you see my screen? Can somebody just tell me yes or no? You guys can see it. Okay, so I just launched eSword, right? So it's eSword.net, you open it up like this, right? So now the advantage that we have here, let me just, online students, you can hear me and you can see my screen, eSword. Faster, all I can see is your picture. You can only see my picture. Yeah. Okay, let me pause here. Let me do the sharing again. Okay, you're seeing that. Let me share my screen and then switch to eSword. Oops, not that, not that, sorry. Switch to eSword. Can you see eSword now? No, Pastor. I'm going to try it again. All right, let me do a window. Okay. All right. So now you can see eSword now. Can you all see eSword coming up on your screen? They're saying anything here. Can somebody unmute your mic and just say yes or no? Yes, Pastor, can see. You can? Yes, yes. Okay, great. Thank you. All right. So this is the tool. Like we said, you can download it free from eSword.net. They have a free version for Windows, but I think for Mac, they have a small page. It's a small fee. So you can download for free. Put it on your laptop, your computer. Now it has lots of things built into it, right? So here you see all the books of the Bible, right? And there are free versions of the... Now you see all these are different versions of the Bible, a literal standard version, Modern King James Message Bible, New American Standard. It's from 95, New American Standard, which is from 2020, New King James, and so on. So a lot of these versions are free. Okay. You can download them into your eSword. Now the King James Version, when you say King James Version Plus, it comes with what is referred to as these Strong's numbers. That means these are the Greek words. So in English, we have the word likewise. If you do your mouse here, that's the Greek word. All right. So we have what is called as Strong's Concordance. It gives you a number for every English word. There's a corresponding number. And then that goes to the Greek word. So for example, likewise, if I put my mouse here, you can see the Greek word. Yeah. You can see the Greek word and you can see the meaning. It gives you the grammar and the meaning of it. You can't see it very clearly, right? On the wall. I'm going to sharpen the focus here. Just if you can sharpen the projectors focus. All right. So if I, you know, for example, I want to know what is the meaning of the word Subjection, right? So in English, King James is the word Subjection. If I click on this, I can go to Strong's and I can look at the meaning. Yeah. That's the best stuff. Okay. That's fine. So I can look at the meaning of this word. So Subjection. The Greek word is Hippo Tasso. And then what is the meaning to subordinate to obey to be under and so on. It gives me all the meanings of it. And then we also have dictionaries. So here is one dictionary. It's called Thayer's dictionary.  And you have other dictionaries, you know, for example, you have mounts. It gives you the meaning of that word and where the same word is used in other places in the New Testament. Okay. So like that, you have dictionaries here and many of these dictionaries are free. You don't have to pay for it. Among the best dictionaries is the wine's dictionary. Right. So the wine, wine's dictionary is a paid version. So you get a paid version of the wine's dictionary, wine's New Testament, wine's Old Testament, and that you can search for words. I'll show you a little later. But are you all following me or are you lost? Okay. So basically, you know, for the English word Subjection, I'm showing you how you can look up the Greek word and how you can get the Greek meaning of that word, you know, so you can look at it. The same thing you can do for Hebrew.  Now notice these are all Hebrew words.  What is the word blessing? So you can do your mouse over here. You will see this, the word blessing means benediction. It means a pool, prosperity. So it means that you can also go here to Strong's and you can look up the Hebrew word and he'll give you the meaning. You can see that Brown, he has, he also has the Brown's dictionary gives you the meaning of that word blessing. Right. So you can study very carefully. You can study the verse, you know, you can study each word, look it up in Hebrew, look it up in Greek. You can study it. So when you study like that, then when you speak, you have a lot of confidence because you know, I have checked the Hebrew, I know what I'm saying. I'm not just speaking and I have studied it. I've looked up the Hebrew, I've looked up the Greek. I know what it means and you can speak with a lot of confidence because you've actually studied that verse. Right. So this is one way to think. So the advantage is you have so many Bibles, versions of the Bible available. Right. Some are paid, but most of them are free. You can just download and keep adding it. Yeah. Okay. Now I remember before 2000, before the year 2000, when I used to study, I would be sitting at a table like this, then I'll have different Bibles in front of me. So I'll have my New King James Bible. I'll have Amplified Bible. I will have another NIV version, another Good News Bible. You know, I'll have four or five versions, physical Bibles. Right. And then if I want to study a verse, I'll have to physically open it up in all these. And then this Concordance was a big book. It was like a thick book. I still have it. Or maybe I put it in the library. It was a physical book, thick book. I have to manually turn, go and search for the word and look at the meaning. And Wine's New Testament dictionary was a book. Right. So I had to have it here. So to study one verse or one word, you have to open so many books to study. And you have to carry it. You can't move it around. It's there on your table all the time. But that's how you had to study before. But once they released this software, it makes so easy. All you have to do is move a mouse. Right. So little energy. Just move your mouse, click a few tabs and you can see everything so fast. It's made, you know, it's made a big difference. You know, shifting. Now I have those old books that I still have the Strong's Concordance, the Wine's Dictionary, all those books that I used to use. I think I put it out in the library. But that's a big difference. Okay. That's how we used to study then. Now the software makes it so easy. And so you have the chapters, the verses, the scripture here. Now, very easy. If I want to know example, let's suppose I go to Proverbs and Proverbs chapter 12 verse one. Right. If I want to read the same verse in different versions, all I have to do is click on this compare tab. One click. And I can see this verse in all these versions, including Hindi. You can read in Hindi, you can read in different languages if you want. I just put it in Hindi thought I'll learn a little bit. I don't use it much. But you see all different versions and you can compare. Right. There is a lot of advantage in reading it in different versions. You can see how that same verse is being communicated in different versions. See all it takes is one tab. Click compare and you can read that same verse and so many different versions of the Bible. Right. Now imagine right here in this simple tab, I have maybe about 15 different versions of the Bible. Imagine physically having 15 different copies of the Bible and trying to turn into that same verse 15 different Bible. It takes so much time. Here you just click on the tab. You can see it's done. Right. So it makes studying so easy. All right. Then you have a lot of other tools. For instance, it has a search function. So I want to search. So this is how it's useful to do a word study. Right. So you can use a search function. And let's say I want to search for the word faith. I type the word faith and I say in which version of the Bible. So I want to search it in New King James. I can search from Genesis to Revelation and I can search. And by just giving a click, I can see the word faith appears 229 in 229 verses in the Bible. Right. And it's giving me all the verses where the word faith is found. Just one tap. One click. Sorry. And there it is. So I can actually study the word faith. I can just go through from Old Testament into New Testament where the word faith is. It makes it so easy. You got it? Huh? I'm sorry. What? It's not coming up there. It got stuck. Okay. Let me. Is it coming here on the screen? Oh yeah. You can see it here. It's got stuck over there. Online students. Is it stuck for all of you? Like is a screen frozen? Are you with me? Okay. You're with me so far. Okay. So let's continue. Since I am sharing my screen, I may not be able to read what you type in the chat. So just, you know, unmute your mic and, you know, if you keep me informed in case you can't see anything, then it'll help. All right. So let me go back to sharing my sharing eSword. And if you can just let me know. Okay. So we're back in eSword. I just wanted to show the search capabilities. Right. So you click on the search. You search for the word faith and you can find everywhere. Right. Now you can also do partial search. Partial math. That means faith. For example, faith, the word faithful. Right. So if you want to include other, the full forms, like faithful, so you do click partial match and you search again. It'll also include, you know, other places like faithful, unfaithful, so on. So that 375 versus found. So you can study like that. You know. So if you want to search only in a book, in one book, you can search. Or the pop-up window, you're not able to see the pop-up window. Okay. Oh, it's not showing. It's showing here. It's not showing on the online. No. Okay. Let me, let me see. Okay. Oh, because that opened the pop-up window opens. It opens a new window. Yeah. I will share my entire screen. Share my entire screen now. Can you see the eSword? Okay. You can see a sword. Right. Okay. Let me. Yeah. Okay. Now you can see. Thank you. Thank you. All right. So what I was saying earlier was, you know, we could search for the word faith and we'll be able to see in almost, you know, an instant. You can see everywhere in the Bible where the word faith is being used. Thank you. Thanks. Right. You can also, you know, do a partial search, which means you'll find words faithful and so on. So you do a search again. You'll find, you know, where would faithful, wherever faith is used, faith and faithful and so on. So you can do that. You can search within books. You can search within, you know, Old Testament, New Testament. So you could, it's very useful doing this search. Okay. Similarly, the wine's dictionary, wine's New Testament, you can search within the wine's the dictionary. So this is where if you search within the dictionary, so I'm searching within wine's dictionary and I say, you know, I want to know the word faith in the dictionary. Yeah. So of course it gives me all the words here and I can go to faith. Now, so wine's dictionary faith, he tells me now. So this is like, almost like he's done a word study for you. You can see this on the thing. Yeah. Thank you. So he shows, you know, faith. The Greek word is pistis. And then he says, this is what it means. And he shows you, he has all, you know, where it is used in the Bible. How, how is the word used? Right. So like this, for example, let's look at the word love and the New Testament. So we click on the word love in wine's New Testament. So he says it's used both as a noun and as a verb, as a verb, agapo. So then he gives you the meaning of the word agapo. And then, but there are other words, Greek words used for love. Philio, right? And he tells you Philio is different from agapo or agape. Philio is referring to affection. Then there are the noun words. Agape is a noun. Philanthropia is another word, noun. And Philadelphia is brotherly love. So basically, you know, that one word, English word love, you have different verbs. And different nouns that are used in the Greek. And so this is a great way to study, you know, and also to see different places where that word is used. The other thing that we have in this tool, and there's a lot, a lot of things, but the other useful thing are commentaries. Okay, now there are literally many commentaries available freely, right? And it's just not possible to read all the commentaries. And I don't necessarily read all the commentaries, but the ones that I would recommend, one is David Gozik's commentary. The reason I recommend David Gozik is because he comes from a spirit-filled perspective. Right? So he acknowledges the work of the Holy Spirit and he comes from a spirit-filled perspective. So that's one commentary I would recommend. Finis Dek, this is a purchased commentary. So you have to pay for this. David Gozik is free. You can download it. Finis Dek is a paid commentary, but that's also something I use. And then, of course, that Vincent word study. So when you are studying a word, he'll give you the meaning of that word. Let's look at an example. A virtuous woman. Okay, Finis Dek, he gives you some... He's studying on the word virtuous and there are cross-references also. Where is this word being used? Let me find another word. Maybe righteous. Okay, now, Vincent word studies. Maybe he does only the New Testament. Right? Yeah. So Vincent word studies. So he comes up here, he gives you commentaries on words that are used in Mark chapter 14 verse 1. Suppose you go to Mark chapter 14 verse 3. He'll give you some commentary on these words here. And David Gozik has this commentary here on Mark chapter 14. So he has a chapter by chapter commentary. So there's a lot of information here, but this is useful. I don't always read everything. Sometimes when I want to... If I want to get some more information, I would read these commentaries, but it's a useful thing to have. Now, the thing is there are many commentaries. Okay, so don't go there and download everything because you won't read it. You won't have time to read it. So download a few that you will actually read and it will help you in your study. These are things that I would recommend. My cross-reference Vincent word studies that you can use. So here when you go to download, that's where you can download various Bibles, commentaries, dictionaries and so on.  All these different Bibles are available. And it's more than what we can actually use, but all these Bibles, this whole, all of these are free. And then there are those that you can purchase like the amplified Bible and so on. And they're not very expensive. And there are, again, free Bibles in different languages. If you want to read in German or French or other languages, you can read. Okay, similarly commentaries. Look at it. If you click on the commentaries tab, you see all these commentaries are available. Now I haven't put all of them here, but there's so many commentaries that are available. You can put them down, you know, and then they're also purchased commentaries if you want. There are dictionaries. Again, there are so many dictionaries available for free. And there are dictionaries that you can pay for. Reference books are also there by, you know, old authors, people from, these are all public domain books. So you could, they're giving them for free. So a lot of information available. And honestly, we don't have time to use all of that, right? But you just take what you want. But the thing that I find most useful is looking up the Hebrew and the Greek. That's what I do most of the time, almost all the time. And I'm studying.  So I just make sure I understand the correct meaning and then interpret it from that. Okay. Any questions? Telugu, huh? Yeah, I am sure they will have Telugu. Yeah, Bible, Hindi, they have Malayalam, Spanish, Vietnamese, Urdu, Telugu. Ah, Telugu. They have two. They have the easy to read version and they have the Indian revised version. The Indian revised version is from 2019. So I think it'll be most latest. Any other language? Good. See, they have, they have many languages. Hindi. I don't know. We can check. So this is a very good tool. I'd encourage you to FGH. Indi, they have two. Yeah, same thing. Easy to read and Indian revised version, two versions. Okay. So I'd encourage you to use this tool to study the word. Okay. Let me just see if there are any questions from our online students. Any questions from your students online? Yeah, I haven't tried it on the phone. I think they have for the phone. You can check on the website. But I think it's, they might have a paid version for the phone. I'm not very sure. Just check. They have a paid version for Mac. App phones, apps phones. Okay. Yeah. So, yeah, they may have, you know, free versions, paid versions, all that. But this is a very useful tool. Now any questions from our online students? Everything. Okay. Any questions? All right. So let me just share the PDF. I'll make myself visible shortly. So we just looked at Eastward being, this is what, I mean, right now, we just went to Eastward. I'm going to give a quick walkthrough on it. I'd encourage you to download it and use it. And I mentioned some of these things here, the commentary that I would recommend. Are there a lot of other commentaries? But, you know, when you read a commentary, don't just take everything they say, you know, because a lot of it is their own personal thought, personal, I think. So, some things you can take, some things you don't take, just leave it aside. And there are useful dictionaries, the Wines Dictionary and the Dix Bible, all of these things are there. Now, along with this, there are many other resources. Okay. That you can go and explore. And many of them are free. There is the U version that, you know, most of us have it on the phone. We have many versions of the Bible, audio Bible, so that's available. There is Blue Letter Bible. Again, a lot of commentaries are available in the Blue Letter Bible. Again, in my observation, there is so much information. It's almost like overwhelming, right? But thank God, you know, they've made all this available for free. So, some people like to use Blue Letter Bible. It's just another online Bible study resource, right? You can go there and they have lots of commentaries and I think they've built in sermons, sermon clips so you can actually listen to what some so-and-so said on that passage. So a lot of things have been built in. Another major one is from Logos. So you can go to Logos.com and download their software. I think I did download it as well. But here again, this particular Bible software is very rich, meaning it has a lot of content in it. A lot of modern or contemporary teachers and authors and all of that. So Logos Bible software has a lot of that content. Again, it's loaded. It has a lot of content information for Bible study and so on. And BibleGateway.com, that's online as well. I think some of you may be familiar with it. That also has a lot of resources. So I have spent most of the time on eSword. That's because I use that. But there are many other tools available today that you can use. You can have a look at all of these. You see whatever you're comfortable and you use it. But I just want to mention that keep in mind that there are theologians from different perspectives. So for example, Logos or Blue Letter Bible or so on. There are Bible teachers and theologians who come from many different perspectives. That means they will, for example, about the baptism of the Holy Spirit. So if you listen to somebody who is from an evangelical perspective, they may give their view on it or read what they said. They'll give their view on it. You look from somebody from a Pentecostal perspective. They will give their view on it. So you have to be a little careful because you'll get confused. Because on one verse, if you have so many different Bible teachers talking and commentaries, you'll find differences. So that's where you be fully persuaded. So you say, I have studied the Bible. I am fully persuaded on the gifts of the Holy Spirit. Or I am fully persuaded that, yeah, today we can speak in tongues and so on and so forth.  Because you'll have all different speaker authors and Bible teachers coming from their own denominational perspective. And it can become very confusing. So that's why when I study, I kind of limit myself to reading the Bible, looking at it in different versions, look up the Hebrew, the Greek and the dictionaries. That's basically what I do most of the time. I don't waste my time reading all the different commentaries. I might look a little bit here and there, but that's occasionally. But most of the time I do my study, then I can come to conclusions based on my study. Any questions on this? Yes. Sorry, Prince. Could you repeat that question? Or you're saying, or do we have Hebrew and Greek in the Blue Letter Bible? I am sorry, I had my camera off all this while. Hebrew and Greek in the Blue Letter Bible, I'm not sure. It might be, I don't know. I haven't really checked that part. Yeah, Roshan, you want to say something? Pastor, I guess there is Hebrew and Greek in the Blue Letter Bible because I've used it. I'm sorry, Roshan? There is Hebrew and Greek in the Blue Letter Bible. I have used it. Okay, thank you. Yeah, so Roshan says that the Hebrew and the Greek is there. So he's used it. Thank you, Roshan. Okay, any other questions from anyone? Yeah, Sean. From the others, what would be the second best option? They're all free. Eastwood, the one for Windows is free. The one you download on your laptop, that's free. Among the others, honestly, I haven't used the others extensively. I've just gone and seen it a little bit and seen what it has, but I haven't used them extensively. So I wouldn't be the right person to make a comparison of the others. Yeah, so yeah, I don't think I should say something when I haven't really compared them. I know they are there. I've gone and seen them a little bit and I've had my first impressions of those other free Bible study websites and software, but I haven't really compared them to say that this is better. So yeah, I'll just let you maybe go look at all of them and see which one you feel comfortable with. I think the important thing is to feel comfortable using it. For me, I felt overwhelmed actually. I opened Logos and I have so much information. I said, I don't have time to read all this. It's good the information is there, but it's too much. I don't have time for that. Similarly, the Blue Letter Bible, there's so much content. I don't have time to go through all that. So I like something very simple that helps me do my study without getting distracted here and there with so many other commentaries and so on. So anyway, I will let you go and explore these tools, see what you're comfortable with and use whatever you're comfortable with, but these tools are available. I'm sorry. I've heard of Olive Tree. I haven't used it. I haven't used it. You can take notes on it. I'm sure there are probably others as well, but I haven't used. These are some things I've seen. So the point is this. The point is there are a lot of resources that are available. You pick one and study. The only warning I would give is that don't get too distracted with the information that's around there. Make sure you study the Bible. Otherwise you'll be studying commentaries. You'll be studying what some person said. And your goal is not to know what, you know, so and so said. Your goal is to know what did the Bible say? What did God say in his words? So study the Bible, but use whatever tools that will help you study the Bible. Otherwise what will happen is there's so much of these commentaries and all that. Don't waste your time reading the commentaries, reading, listening to this person, that person.  Right. To help you do it if you want to read a little bit outside here and there. Okay. But don't get caught up in it because there's so much information. Yeah. Right. I'm going to. Yeah. So I'm going to move forward now and maybe just introduce our next lesson. Any questions? Online students, everybody's very quiet here. Okay. All right. I'm going to move forward to the next let me share. So having gone through the tools that are available for us, what I want to just not sure what happened here is that as we progress now into biblical interpretation, it's very important to depend on the illumination of the Holy Spirit. So on the one side, we are going to do our parts. Right. You make use of the tools, look up the Hebrew, look up the Greek. You read it in different versions, read the dictionary. Okay. Okay. All that is good. That is your part. But we have to depend on the Holy Spirit. So we have to be able to bring this together. Right.  Say, Holy Spirit, help me understand. And our goal is to receive the illumination that comes from the Holy Spirit, because he is the one who inspired the Scriptures. He is the original author of the Bible. He used men to write it. Today, here we are, we are doing our study of, you know, doing a background, doing a cultural study, this, that, all that we're doing. That's our part. But then we have to depend on the Holy Spirit to eliminate my mind. He will give us insight, will give us revelation. Okay. So that is important. Now, what we know is that the natural man cannot understand the things of God. That means the man who is only depending on his mind, his intellectual power. Right. Spiritually, he's blinded. And the natural man, the unsaved man cannot understand spiritual things. Okay. So we must not depend exclusively on our natural mind. Use the mind. Try to understand the Hebrew, the Greek, the English, the language, etc. But depend on the illumination of the Holy Spirit to give us insight into Scripture. And so, you know, in Ephesians chapter 1 verses 15 to 21, the Apostle Paul, he prays for the Ephesians. He says, I'm praying that God will give you the spirit of wisdom and revelation. Right. Understanding revelation in the knowledge of him. Right. So the Holy Spirit is the one who gives us wisdom and understanding in knowing him, in knowing our calling, in knowing the inheritance that is there, in knowing the power of God. The Holy Spirit gives us wisdom and revelation. He has to open our eyes, enlighten the eyes of our understanding and give it to us. So just a thought here, inspiration was given once. The Holy Spirit inspired the authors. But illumination is given over and over again. That means he gives you light over to all of us, different people. As we read, he illuminates. He opens our eyes or he gives us revelation. Right. And he helps us see things that we need to see. Right. Now, I want to just point to something here. And we have this great, beautiful example in Matthew 16, verses 13 to 23. When Jesus asked his disciples, whom do you say I am? Whom do people say I am? So they said, some people are saying John the Baptist, some people are saying Elijah, so on and so forth. Then Jesus asked his disciples, whom do you say I am? You can imagine. They've been following Jesus for almost two years, about two years. I don't know exactly how much. They've been following him. They have left everything. They are literally living with Jesus. And now he's asking them, who do you think I am? And they're all very quiet. You're following this person. You've given up everything to be with this person. And you don't know who he is. They're quiet. It's a very peculiar situation. Then suddenly Peter speaks out. He says, you are the Christ, the son of the living God. You are the Christ. You are the Messiah. Christ means Messiah. You are the Messiah, the son of the living God. And then Jesus tells Peter, Peter, flesh and blood has not revealed it to you, but my Father in heaven. That means Peter, you did not learn this from other people.  So how did Peter, how did the other disciples know who Jesus was? How did they know? Who was Jesus? If you look at it in the natural, Jesus was just an ordinary person. He was born in a carpenter family. He must have been working as a carpenter for 30 years till his age 30.  He's hitting the hammer and doing all that. Look at him carpenter. But now these disciples, in their heart, they know this is the Christ, the Messiah, the son of the living. How did they know? Jesus, Peter, flesh and blood didn't reveal to you, but my Father who is in heaven. So see, this is revelation. It is something you know inside you, which God has revealed to you. And sometimes the natural doesn't agree or they may not be too much natural evidence because you look at Jesus, he's just a carpenter. He was born to, in that family, he was working like a carpenter. His father is a carpenter and the natural, that's what it is. But they fell. Peter, you are Christ, the son of the living. How Peter, how do you know this revelation? Inside me, I know it's like my eyes are seeing something. My spiritual eyes are seeing something that my natural, that is not very obvious to my natural eyes. That is revelation. You understand it? So we need that. Paul prayed for believers. He said, may God give you the spirit of wisdom and understanding to see these things. But I also want to point out in that same chapter in Matthew 16, the same man who received revelation from the Father also bent into error. So what do you mean? In that same chapter, afterwards Jesus was telling his disciples, I'm going to be crucified. I'm going to be delivered to the hands of the Gentiles. They're going to crucify me. And then Peter says, Lord, that is not going to happen to you. And how does Jesus respond? Get behind me, Satan. Five minutes back, the Father has given you revelation. Five minutes later, get behind me, Satan. I'm just saying five minutes. I don't know exactly. We don't know exactly how much time I'm just making up. But shortly after, get behind me, same man. Get behind me, Satan. You don't understand the things of God. A few minutes back, by revelation, he was saying, you are the Christ, the son of the living God. Some shortly after, no revelation, full confusion. He said, I won't let you go.  But the point is this. The same person can be right about one thing and wrong about another thing. Same person. He can be right about one thing because God has revealed that. But he can be wrong about another thing because he's operating from his mind, his own mind. Same person. So we have to be very careful. Okay. We'll pick up here next class. I heard the bell ring. So we'll stop here. I will get ready for the class next class and we'll pick this up. Okay. All right. Let's pause here. I'm sorry for stopping abruptly. We'll get ready for next class. Okay. Thank you, everyone. We'll pick this up next week and take this forward. Thank you. Bye now.


Yeah, cause last week we were, you had to juggle a lot, saw a lot of different stuff, a lot of different guys out there including Darren making his first NFL start. What was that process like?  You know, we tried to, for the first time all year, match up AJ on digs. And as I've told you in the past, that's not always the easiest thing to do. You know, and we struggled with it sometimes.  We had an uncommonly number of missed tackles, which killed us in a run game. We had 15 of them and we haven't had 15 in three games usually. We had 11 missed assignments and just all that a little bit because we were trying to change some things around to try to get the best matchups we could. But when you try to do that, it's not the norm. It's not what you practice for a year. You try to do it thinking it's going to give you the best chance. I don't think it necessarily did. You know, they didn't hurt us in the passing game. We got picks. We got some things. We did some good things there. It really hurt us in the run game. We ended up with some bad run force situations on the edge. And that contributed sometimes to the guys being out of position, which gave them bad angles, which made them miss tackles. It's just, it wasn't, you know, I've said it in the past. I don't like to do it. And I wish I hadn't done it even in that one. Well, we did well in the past game. The matchup, AJ, on Diggs wasn't a problem because I don't know how many catches he had, but it never felt like that was ever a problem. I mean, he caught a couple. But the problem with it is when you do that, then somebody else has to figure out who they got. You know, you got two other DBs, there are corners in there, a nickel and a corner, and they're young. You know, we had Sheffield who's playing for the first time, then you got Darren starting, you know, and Sean is back there and a lot of new guys. And even though we thought it was maybe the best way to do it in retrospect, I don't know if it was, because what happened was we just, it wasn't the passing game. It was a running game that ended up hurting us because we had some guys get lined up late and meanwhile the ball's being snapped and run around the edge and we don't have anybody there to help the defensive end. You know, whenever you try to help somebody, you're hurting somebody, you're putting another load on somebody else and they got a good running scheme, got a really good running scheme, a really good running game. And a quarterback really hurt us. And that was something that we, that was the biggest emphasis going into the game was take care of the quarterback and we didn't do it. We did it in the passing game. We didn't do it in the running game. Had chances of sacking him on a 3rd and 8th, you know, to get off the field on the first score. We should be off at least, you know, hold him to a field goal. We didn't, we missed a tackle. It just, you know, just disappointed in the running game, disappointed in myself, just thinking through, just saying, I know better. And I tried to do it and it didn't work very well. What do you think Darren handled in his first start most of the day? Okay.    But I'm just saying that, you know, he did okay. He did okay. And for him and all the rookies in the defense, how do you think they've handled their first season adjusting to what you guys have been doing for a minute? I'm pleased with them. Actually, they've made some mistakes like, like every player has. And they're typical rookie mistakes.  But we have made a lot of growth. So I'm expecting going into next year that that growth will really pay off for us.  They've been through it. And so hopefully, you know, starting out next year, it's not going to be like, you know, brand new to them. There are going to be a lot of carryover. And hopefully with COVID ever gets gone, we'll have a spring and all that kind of stuff and meet with them and do all that kind of stuff that it'll pay off in the long run. It'll pay off. I'm pleased overall with all of them. You talk about that growth and this will probably be the last chance that we get to talk to you as the season winds down. I mean, how, I know you talk about them understanding the scheme, but in what ways have you seen this defense evolves from what you maybe thought this defense would be in training camp early in the season and now what it is today? Oh, I've seen big strides. You know, we're on a 12 game streak of having a turnover in every game in the last 12 games. You know, the thing that we just got to get better, I've seen tremendous strides from early in the season where it seemed like pretty much whatever you called, there was one guy making a mistake somewhere. Whereas now there's a lot of calls and nobody's making a mistake. You know, we're making some plays. The thing that we just really got to develop in the offseason and before next year is the ability to rush the passer and not always have to pressure to do it. And number two is to be able to play man coverage better in the back end and because it's kind of a two fold thing. If you don't get to the quarterback, that means the secondaries hung out there forever and that's not good. On the other hand, if you can't play man coverage at some point in time in this league, I mean, you're just going to have to. So it's those two things to me that we really have to develop before next year. It's not now learning the system.  Those are the two things that just have got to get better next season. Do you develop those things with experience or do you have to do it through personnel? Both, all things. You know, I just, I can go back to so many players. I can watch so many players on film now watching. I watched guys rush the passer and I'll guarantee you the coach didn't coach them. The guys got talent. I mean, we did a lot of things. We did a lot of things.  It wasn't exactly coaching, you know, didn't make Ray Lewis, didn't make Terrell Suggs. Those guys made themselves. We just, I didn't screw them up, but it's just, it's both. They just, we got to develop the talent that we have, whether it changes, you know, every team changes in the off season. Like I told the players yesterday or in our meeting actually today, you know, this won't be the same team next year that's sitting out here right now. It never is. I don't care whether you win the Super Bowl, you have a good season, bad season, average season, doesn't really matter. No team, I think the Bucks are about the only one I've ever heard of, Tampa Bay, that actually got everybody back. Here we won the Super Bowl in 2012. We lost seven players off that defense. It's starters. I mean, they all got contracts and two of them retired. So it's, you know, it's just, it's going to be with probably different talent and it's also going to be us doing a better job of coaching and getting them to do the things we need to do. I was watching Grady Jarrett's mic'd up from a few weeks ago and there was one moment where there were three guys on him and he said, damn, you can't block anybody else or you don't want to block anybody else. And that actually, there really was a play a couple of weeks ago in Detroit. I think it was Detroit, right? It's like, I was the thing up. They got somebody ought to be free. I mean, there's three guys on him. We actually did make a decent play on it though. So if they block three of them and leave somebody go, that's fine by me. As a coordinator, when you have a guy like Grady Jarrett who is getting double teamed at the rate of what he's being double teamed, is there anything that you can do to free him up or is just one of those things where it's like the respect that he has in the league, people know what he brings to the table? We've tried. I mean, if you notice, there's been some times we've played him outside on some things and try to rush him from outside and try to keep him away from the big guys inside. But it's kind of inherent to his position that you play in there and you're good. They're going to double team you. And the thing that you got to be able to do is be able to put somebody usually next to them that they got to also worry about. It's like anything. You know, if you play beside Aaron and Donald, you got a good chance you're going to end up one-on-one. If you play beside Suggs, good chance they're turning the protection that way. So if you're the other end, you know, then you got a pretty good chance getting one-on-one. I mean, one year, I can't even remember what it was, 13 or 14, we had Suggs on one side and Elvis Doomerville on the other side. Well, Doomerville ended up with 17 sacks. Guess why?  But the point was is that we just need more. We need to develop past rushers and that'll help Jared if he, all the stress isn't necessarily put on him. And we have tried to do some things to free him up. A lot of our zone pressures are, we put him in different spots so we can try to get him one-on-one. And sometimes it'll work, sometimes it doesn't. You probably haven't had time to say, yes, whether this generation of players quote-unquote loves football. I wonder, you've been doing this for a while, what's your take on the decisions? I'd love to tell you, but it wouldn't be fair. It wouldn't, it would not, just I'd love to tell you, but the truth of it is it wouldn't do me any good to say something because then players are going to take that as either one thing either I like it or I don't. But in either case, I don't think either one of them is going to benefit me as a coach when the players hear this press conference. You know, there's just things when you guys ask things just like asking about players. I'm always thinking about if I'm the player, how would I like to hear my, what would I like to hear my coach say? And if I'm a young generation player, the last thing I want to hear is some coach say that we're a bunch of softies or they weren't this or that, I don't believe that. I think it's all individual. I think it's, hey, there's some guys that are and there's some guys that aren't. There's some guys that are motivated by just the game of football and there's some guys that are motivated by money and there's some guys that are motivated, I don't know what everybody's motivation is. There's coaches too. Now I'll tell you about the coaches. Because I think the younger generation of coaches feel a little entitled. Yeah, I think they're spoiled. I think they, hey, go work in a high school, go work in a Division III school where you got to mow the grass, you got to align the field, you got to do all those things and you'll appreciate what you have when you have it. Instead of being 25 years old and wondering why I'm not a coordinator already in the NFL. Okay, I went to the NFL at 55 years old. I was a high school coach, I was a Division III coach, I was in the Mac as a coach. I didn't go to New England until I was 55 years old. And so I felt like I paid my dues and I feel like it made me a better coach, made me a better teacher. I was a school teacher, I learned how to teach, not just stand out. I look at guys now and they can't stand up in front of the room and talk to people. They can't, they got to get on the computer. Everything's computerized, it's still a people's game. Players want to talk to you, they want to hear from you. It's not that, I still do everything in writing. I don't do all my own breakdowns. I don't ask some quality control guy to do it. Everybody gets on a computer for two years and thinks they ought to be a coach. It's not mad in football. It has to deal with people. It's like all the analytics there, everybody talks about this and that and you should do this, you should do that. The computer told you that. When did the computer know what the weather was, whether it was raining, whether the wind was blowing, whether you were playing good on defense? Okay, they say, well, it's a two-point game, should you go for it on 4th down? Well, I don't know, is it score 42 to 40 or 6 to 3? It makes a difference. If you're playing great defense, then maybe they're playing great defense, don't. If they're playing a lousy defense, yes. Peter doesn't tell you that. So it's kind of that way to me in coaching. Players want to be coached. They do. They do. All of them want to be coached. They want to be good. They want to be coached. They want to be told what to do and how to do it and correct them and talk to them and be honest with them. And I just don't feel in this generation sometimes the coaches that they have very good personal relationships with players. I love my players. I always love my players. Everybody asks me, who's your favorite player? I go, all of them. All of them. Anybody that played for me is my favorite player. So to me, that's what you learn over the course of 48 seasons is, you know, to just go coach them and be personal with them and care about them. You know, and I just, anymore, it's how fast can I climb the ladder? I didn't climb up very fast, but I feel good about the way I did it. So long story short. We like long stories. That's all right. Anything else? I don't want to get all deep on you here, but now that you've been back in the game for a full season, have you enjoyed it? Oh, absolutely. Love it. Love Arthur, staff, players. I appreciate these players so much. They are given everything they got. They really are. And what else can you ask of somebody?  And these guys give us what they got. And so do the coaches. I love the staff. This is an older staff now, you know, in defense. And I like it that way because I know how they've come up and how all that stuff. And we got some young assistant coaches who are learning from some older coaches, which is a great thing. I mean, Arthur's put together a great staff. And I absolutely loved it. Love the area. Love Atlanta. Love it. You guys good? Yeah. Okay. Thank you. Thank you guys.


Okay, move out. That in city's course in West Berlin. And the ready-fighting men of a reserve brigade. All in this issue of your army reports. The Cheyenne helicopter made its first public flight. Reverse thrust from the thrusting propeller. He's not using brakes. He now will show you some flexibility by reversing again putting the thrusting propeller in the rear in the reverse position. It turns on the ground now. Turning to the left. The rising aircraft has ever flown before. But again demonstrates the high stability and versatile handling qualities of the Cheyenne. Stability is a requirement for the aircraft to put a tremendous firepower on the target. Using the thrusting propeller as he is doing now, he's applying reverse thrust to the propeller and you notice the nose-low attitude that the vehicle attains. This gives it an increase in the aiming angle for the fixed guns and rockets. He can get to 17 degrees nose-low and approximately 20 degrees nose-up using this technique. This is going to be a medium speed run about 100 miles an hour using the pressure propeller and watch her go up. Right up to its service ceiling of 26,000 feet. This will be a relatively low speed pass about 110 miles an hour. Army officials on hand at the Cheyenne's first public flight stated that they were impressed and pleased with the new helicopter's performance. Adding that the Cheyenne will make a valuable addition to the Army's fire support capability. They also stated that the Cheyenne helicopter is scheduled to enter the Army's inventory on all weather, day or night, fire support system in the near future. East German soldiers man sentry posts which keep close watch on American forces training in the western sector, nearby throughout the year while United States troops assigned to West Berlin. East German soldiers observe the training exercises. Although the ammunition used is harmless, the training is otherwise carried out in the most realistic combat manner during these simulated attacks. One of the troops in the mock battles, as in the real thing, is to close with and kill or capture the enemy who are blamed by other U.S. soldiers acting with aggressiveness from its surroundings by ceasing key positions, overtaking enemy fortifications. Managed points overlooking fields of fire are important objectives during the exercise for added realistic effect. Limited mobility and restricted communications within built-up areas places increased responsibility on the initiative of a small unit leader whose judgment must be relied upon. Grappling hooks with ropes and scaling ladders are methods used to enter higher floors of a building. Pieces of white cloth are thrown over the windowsill as a signal for shockwave troops to advance. To move out to conduct a house-to-house search, flushing the aggressor forces hiding within these buildings points overlooking sniper positions. To securing one building, the troops move on to the next under protective fire cover using their house-to-house search of the mock city. The exercise comes to a successful conclusion with its many valuable lessons learned. Within view of East German sentry posts, continuous realistic training of the U.S. soldier in the combat in cities course in West Berlin ensures that our forces will remain on the alert, combat ready to meet and defeat any act of aggression, any time and at any place. 100 Fifth Brigade arrives at Capicoy from 28 cities in Minnesota, Iowa and Wisconsin. Its units range from the oldest in the United States Army to the newest. The old guard is here and the 14th artillery, the 17th infantry, the 409th and the 410th. The 33rd armor and the 4th cab are represented. There are engineers to build bridges, search-like companies to light them, smoke companies to conceal them. There are military police, military intelligence and military public information units. An aviation company to fly to choppers and the gunships. A support battalion to link them all together. The brigade brings with it the battle-hardened veteran, the decorated officer, the army-wise sergeant and the raw recruits. The immediate mission will be to weld these units and these men into a cohesive, mobile, self-supported striking force. To one man, the mission is paramount. For he must prove his unit's worth. Barber, Arnold T. Brigadier General, commanding. The mission of this brigade is to move from our assembly area at this point and move into a mobile defense with preparation of launching attack for the intelligence. We have this information at this time. We have two mechanized battalions to our front. They're located generally in these two areas. We have two tank battalions that have been reported moving down this highway here. Lights burn through the night. Orders are on the way. The mission, Jack Pine 1. In the war room at core headquarters, each unit of the brigade becomes a box or circle or flag or pin on the huge situation map. Radio's crackle-urgent command nerves tingle. The pulse pounds. The brigade brings. The garage will go to the field. It will turn twisted method into functional parts. In the motor pool, routine maintenance is now top priority. Emergency aid packs, plasma, bandages, information tags, and medications are moving down a hasty production line in the medical company. The materiel, the men, the machine are moving on the mission. Field rations, generators, communications wires, and crated ammunition. Reaching a rendezvous at dawn. Second platoon, Charlie Company. Fourth battalion, 33rd armor. Commanded by Lieutenant Dale John Worth. A 26-year-old officer trained at Fort Knox and rising through the ranks to a commission. Like many armor officers or tank commanders, Worth teaches tactics, which take advantage of the tank's mobility, firepower, and brawn. A tank has actual and psychological powers. Worth will use all of them in his mission. The artillery, speed, and power. One, one, zero. Bravo site, plus four. Bravo site, plus four. Deflection, two, seven, three, two. Quadrant, two, eight, seven. Center one round, battery one round in effect. Back instantly. Crews receive a mission. Pull off the road. Train the howitzers on a target, perhaps seven miles away. Umpires and evaluators mark score sheets, leading to the pressure. Three rounds will be fired for record. The air attacks have finished. The shelling has died away. The infantry will take the high ground. The infantry waits, wet, warm, weary, in the midst of dawn. The infantry feels its belly churn in the moments before attack. The infantry moves on foot in the driving rain or in the treacherous quiet of the sun-dappled wood.  He carries his home upon his back, marches through three pair of boots, lives off the land, sleeps where he can, eats from a brown box, labeled meal combat. He is bare target from the concealed bunker, the long-range guns, the mortars, and from the sky behind the hill. The separate infantry brigade has infinite capabilities. It can launch an armor spearhead, command artillery, feed and clothe itself, establish its own communications, and through its powerful generators light an entire city. Combat engineers build bridges of every description. Their bridges carry tanks, trucks, equipment, supplies, and men. The infantry is on the move. Raw firepower is not enough. It must be mobile. The 106 millimeter recoilless rifle meets such requirements. It is jeep mounted. It packs an armor piercing punch from 8,500 yards. The weapon is a massive 4.2 inch mortar, mounted on the carrier, or fired from the ground. It will hurl its shell 6,000 yards with pinpoint accuracy. Number two, deflection. 0, 7, 9, or 5. In its role as a self-supporting force, the brigade must care for itself on the attack, on the defense, and in grim moments of tragedy. It replies with intensively trained professionals in its medical company. It responds to the urgent call with ambulance and helicopter. Time is precious. The skilled corpsman will waste none of it. Their techniques are far advanced from those used in World War II and in Korea. And to the rear, the surgeons are ready. They will complete the business of saving lives. Pity and Burley, he commands a recon platoon. Shiblin, William J, first lieutenant, third infantry, 26, Ranger, paratrooper. His specialty is an old one, forged behind rock walls and hedge roads, sharpened in countless gullies, and along a thousand dusty roads, home in the jungles of Vietnam. Bill Schiebler's specialty is the quick skill. Ambush. The citizen soldiers are going home, and with them ratings which range from excellent to superior. The combat veterans, the decorated officers, the army-wise sergeants, and the raw recruits are moving with a purpose. Conquered bridge. They carry on the rich and hard-won traditions of Breeds Hill and Red Beach One, of cow pens and bellow wood, of Utah and Hill 609, of the Han and of the Rhine. And months that follow, they will continue to train. For if they are called again, they must be ready. These men of the brigade.