 Hello and welcome to the first lecture on force models for ascent mission. As we have seen there are three forces that we need to consider that is propulsive, the gravity and the aerodynamic forces. So, let us start our discussion with the propulsive force which is also commonly called thrust. We all know that thrust of a rocket engine is based on Newton's third law that is action equal to reaction. Of course, we can also derive this from the conservation of momentum as per the following interpretation. So, let us create a scenario which says that a rocket engine burns the propellant and resulting hot gases are ejected in the opposite direction at a very high speed. The momentum so generated creates equal and opposite momentum on the rocket so that the net momentum remains constant. But the momentum lost due to the exhaust gases appears in the form of momentum gained by the rocket or as a force on the rocket. Of course, we need to note here that accurate thrust models will require detailed thermodynamic laws internal flow models etc. However, for the ascent mission sizing and initial design a very very basic thrust model can be derived from the concept of rocket engine firing on a test bed which is a common configuration in which most of the rocket engines are tested before they are put on rocket. So, let us look at this particular representation of a rocket engine firing. So, consider the following schematic in which a rocket is burning propellant and ejecting hot gases. Of course, in this case we have introduced a velocity vector V just to make the derivation more general. But we could very easily make this velocity equal to 0 which would represent the test bed configuration. So, we take a rocket of mass m moving with the velocity V as shown in the picture. Let us assume that this is a snapshot at a time instant t and then look at the snapshot at time t plus dt. So, in that time interval dt, the propellant has been burnt by a small amount represented by dm which is ejected related to the rocket at a high speed V subscript E. And in the process, the rocket velocity increases by an amount d as the ejected mass and the rocket are considered to be part of the same control volume. We can say that the momentum within the control volume is conserved indirectly that means that the momentum lost by the rocket through the exhaust gases must be balanced by the momentum gained by the rocket in a net sense. And this can be stated as saying that rocket loses a mass dm with a relative velocity minus Ve and gains a net forward velocity dV. We can write the momentum balance as m minus dm into V plus dV plus dm into V minus Ve equal to the original momentum mV. Of course, we just cancel the terms and ignore the higher order term dm into dV with the net result that now we say that the net change in momentum is 0 that is m into dV. The change in the momentum of the rocket is balanced by the momentum gained by the exhaust gases in the opposite direction that is dm into Ve. Because this change in momentum results in a force in the forward direction, we hypothesize the presence of a force called thrust shown as t in this equation. And then we have an expression for thrust as minus m dot into V. The negative sign here indicates that m dot is a negative quantity because with relative to the rocket we are losing mass. So, rate of change of mass is negative. Of course, on the test bed both V and dV are 0 because the rocket does not move. The net loss of momentum dm into Ve appears as incremental impulse which is like a force. And on the test bed this force is resisted by the support system as a reaction. And typically that reaction is measured as a indicator of the amount of thrust the rocket motor is generating. Of course, the thrust component so obtained is purely due to the exhaust velocity and it also is called the velocity thrust. In addition, there is generally a difference in the static pressure Pe in the atmosphere and the nozzle exit static pressure Pe due to the internal flow characteristics of the exhaust gases and the design of the nozzle. This difference in the pressure also results in a force as shown in the schematic below. So, if you have a rocket in the atmosphere, the atmospheric pressure acts all along it and at all other points it balances each other so that there is no net force in those direction except the front and the back where the pressure at the front is different from the pressure at the back because of exhaust nozzle. And this difference in the pressure results in a net force which can be modeled as a simple differential pressure into area. And the expression for that is as follows. That additional thrust is simply added to the expression for the basic thrust of velocity and we get two terms for thrust. The first one is the velocity thrust and second one is the pressure thrust. Here A is the cross sectional area of the nozzle through which the exhaust gases exit. As I have mentioned, the first term is positive because dm by dt is negative. However, you will note that the second term now can be either positive or negative depending upon the sign of Pe minus Pa. So, we know that if Pe is greater than Pa, you will get a positive value. On the other hand, if Pe is less than Pa, you will get a negative value. This has an important implication for such rocket nozzles which are designed to fire with certain attributes at sea level where atmospheric pressure Pa is high. As the rocket moves through the atmosphere, the static pressure drops until it goes out of the atmosphere so that Pa becomes zero. And we find that if we reach the vacuum condition where Pa is zero, while Pe will be a non-zero quantity, there will be an additional positive thrust available at higher altitudes from the same rocket motor. So, the rocket motors that are designed with a particular thrust at sea level can actually give better performance at higher altitudes. Let us now introduce the concept of specific impulse as a figure of merit for rocket motors. As we have already seen, thrust is a combined effect of propellant and nozzle because the propellant generates the temperature, pressure in the combustion chamber which is then expanded through the nozzle and then this combination generates both exit velocity Ve and exit pressure Pe which are part of our thrust definition. So, this combination of propellant and nozzle is generally treated as a single unit called the rocket motor and is converted into a mechanical figure of merit which is called a specific impulse or ISP. The definition of ISP is as given below. ISP whose units are in seconds is the ratio of the net thrust or the force generated by the rocket motor divided by weight flow rate. So, m dot is the mass flow rate. When we multiply this by g naught the acceleration, this becomes weight flow rate. So, this is the thrust per unit weight flow rate. What it means is that per unit kilogram of fuel burnt in unit time, what is the amount of thrust that one can generate and that becomes a parameter that characterizes a particular propellant rocket motor combination. In this definition g naught is an indicator of the acceleration due to gravity at sea level and a constant value is used as shown in this representation. Of course, as I have mentioned ISP is normally attributed to a rocket motor which is a combination of the propellant and the nozzle and that is why they are also designed and fabricated together as it depends on the characteristic value of the propellant and the nozzle shape. Of course, for different propellants we are going to get different values of these propellants. For solid propellants typically what is used is APCP which is expanded as ammonium perchlorate composite propellant which is essentially a combination of ammonium perchlorate and which is an oxidizer and then you have the fuel which is either HTPB or P-Band. HTPB represents hydroxyl terminated polybutadiene or P-Band which is polybutadiene aniline nitride. Both are hydrocarbons and represent the fuel. This combination is normally augmented with aluminium in the powder form in order to generate very high temperatures because aluminium powder once ignited at high temperature generates a large amount of heat. It is an exothermic reaction and the ISP for such propellants is in the range of 170 to 220 seconds. When we use liquid propellants in rocket motors there are two popular kinds which are commonly used as fuel that is either kerosene or UDMH which expands as unsymmetric dimethyl hydrazine which is the hydrocarbon along with liquid oxygen as the oxidizer or N2O4 as the oxidizer. In these cases we can get specific impulse between the range of 200 to 350 seconds. When we come to cryogenic engine, many of you would have heard that it is liquid hydrogen and liquid oxygen is the most common cryogenic fuel which is used with hydrogen as the fuel and oxygen as the oxidizer. Sometimes like natural gas is also used in place of hydrogen and typical ISPs which are possible with such engines are around 450 seconds. The nuclear propulsion which is also used in some applications of launch vehicles, rocket motors generates the ISP in the range of 300 to 500. With that discussion completed, let us now move over to the model for a gravitational force. If we look at it from a very very basic perspective, the Earth's gravitational model is based on the universal law of gravitation which is defined as follows. So, if we have two masses M1 and M2 placed at a distance of r then they will generate gravitational forces f1 and f2 of the attractive nature given by the expression g into M1 into M2 by r square where g is the universal gravitational constant. This law is universal and is applicable for all bodies including Earth and the launch vehicle. In the context of launch vehicle what we do is that we rewrite this whole relation in a slightly different form to generate a force due to gravity on rocket as given by the above expression minus g M1 M2 by r cube and hypothesize a gravitational potential function which is minus mu by r such that the acceleration due to gravity g which is what we commonly use in most of our calculations is nothing but a gradient of the potential function minus mu by r. Further, for most applications we can make use of a spherically symmetric mass distribution for Earth such that acceleration due to gravity expression as seen above reduces to the following simple expression that is g as a vector quantity is minus mu r by r cube where r is the position vector from the center of Earth to the center of mass of the launch vehicle because r naught here is the radius of Earth in different case. Now, with this basic background on gravitational model we now look at the realistic model of gravitation in the context of Earth. So, the first thing that comes across is that Earth is not spherical and is typically modeled as an ellipsoid as shown below. The blue lines show the ellipsoidal representation of Earth geometry. So, spherical approximation is the red line with an average radius of about 6371 kilometer while in the context of ellipsoidal model the equatorial radius is 6378 kilometers while the polar radius is 6356 kilometers. So, that there is a difference of roughly about 22 kilometers in the two radii in the ellipsoidal configuration. This particular representation has been adopted in defining the gravitational model using WGS 84 standard which is used where WGS stands for World Geographic System. In 1984 this was first proposed and later was improved upon in 1995. So, you have a WGS 95. I suggest that you can look at Wikipedia or any other internet resource to understand what are the implications of the modeling of the gravity using WGS 84 and WGS 95. At the next level there is an EGM 96, EGM standing for Earth Gravitational Model evolved in 1996 where the ellipsoidal assumption is relaxed and Earth is modeled as a geoid or an equipotential surface as shown in this picture courtesy Wikipedia. So, it is the geoid or equipotential surface indicating that the gravitational potential on the surface is same everywhere. So, that depending upon its distance from the center the value of gravitational acceleration different at different locations given by the latitude and the longitudes. This is commonly modeled through a fairly complex mathematical representation which make use of zonal and tesseral harmonics which we will not go into at the moment, but you must be aware of it. The most recent of this kind of model has been given in 2008 under the name EGM 2008. Of course, in the context of our asset mission for the initial sizing of the launch vehicle for most such applications we use the spherically symmetric model to start it and if necessary we can always make the corrections to improve the quality of result. So, when we use this spherically symmetric model we get a gravitational acceleration term, but we do a little bit of tweaking to this to show that we are not going to use a constant gravitational model which means we are going to include the change in the gravitational acceleration due to change in the altitude. This is necessary because you can see from the simple exercise that the gravitational acceleration of 9.81 at sea level or the surface of the earth reduces by 3 percent at an altitude of 100 kilometer and reduces by 12 percent at an altitude of 400 kilometers. As most of the asset mission would end up between 250 to 400 kilometers you will find that as you reach the terminal point the value of gravity will be different from what you would assume at the start of the mission at the sea level. To summarize both thrust and gravity models are simplified versions of more accurate but complex models that also provide better results, but we have also seen that simplified models will help us to quickly size the vehicle and understand its performance which then can be taken for more accurate analysis. So, with that we close this lecture and we will next take up the aerodynamic and the geometric models. So, bye and see you in the next lecture and thank you.