Let me come back to where we were before we dispersed. We defined the work ratio as the second parameter. Now we go into that zone where things tend to become specific. But before that let me say that although many of these parameters or particularly the efficiency is a work, sorry it is a ratio and hence should be expressed only as a fraction or at best or another option is as a percentage. Sometimes it is expressed in terms of certain units. The dimension is still one but the units may be other than one. We will come to some illustrations later. The third parameter is specific output. Now this has something to do with the flow rate of the working fluid. Flow rate or circulation rate working fluid and let us say m dot WF is the this flow rate. Then for power plant the specific output is the net output divided by m dot WF. This is for power plants and the units would be this would typically be watt. This would be kg per second. So watt second per kg or watt per kg per second or kilowatt per kg per second or megawatt per kg per second is a proper unit. But since we know that watt second is joule. So this can also be expressed in terms of say joule per kilogram. For refrigerators and heat pumps for example refrigerators this can be expressed in terms of the specific output may be called specific refrigeration effect. For example this would be called specific power output if you want. Power is the optional word. For refrigeration plant this would be q dot ref divided by m dot working fluid. Units would be the same and this may be called the specific refrigeration effect. And what is the significance of this? You should notice that the larger plants will have larger flow rates of the working fluid. However when you compare two different technologies or two different modifications to a plant it is possible that you will have a plant with two plants with a lower specific output and another plant with a higher specific output. Lower specific output would mean that given a particular flow rate of the working fluid you have a lower power output. A higher specific output means for the same working fluid you have a higher power output. The one with the higher specific output would tend to be a more compact plant. As opposite to this or related to this you can write this as 3A or 4, choice is yours. This would be the essentially its reciprocal. This will be known as the specific flow rate and this is defined as nothing but the reciprocal of our previous specific output. This will be m dot working fluid divided by w dot net. So for power plant this will be m dot working fluid divided by w dot net. For refrigeration plant this will be m dot w h divided by u dot refrigeration. Again the unit, typical unit would be kg per second per watt or kilowatt or it could be kilogram per joule or kilogram per kilo joule because watt second is a joule. Since power is quite often converted into electricity and commercial unit for the sale and transfer of electricity so called simply unit. In fact that is one funny thing. Your electricity bill will say you have consumed 320 units in the last month or last two months that unit is actually a kilowatt hour. Kilowatt hour is the so called unit of electricity transfer or electrical energy consumption. So quite often this will be in terms of kilograms per kilowatt hour where kilowatt hour we know is the so called unit of electricity, electrical power consumed or transferred. Similarly here it could be kg per second per watt or kilo joule, kilogram per joule, kilo joule, mega joule or something like that. These things are more important and more common with power plants than with refrigeration effect. Coming to power plants by the way I should tell you that this thing specific flow rate quite often is known as steam rate or specific steam rate for steam power plant and one should remember that the if this is not rule of thumb but the typical this will be approximately of the order of 1 kg per second per megawatt. This is order that means I do not claim that it will be exactly 1 kg per second but if you see a typical steam power plant which works today with condensing in temperatures around 40 degree C and boiler inlet anywhere from say 50 bar, 60 bar saturated steam to 120 bar or 550 degree C or so. It will be around 1, if you get 0.7 or 1.3 or 1.4 it is okay but if you suddenly get a value of 10 kg per second per megawatt or say 0.1 kg per second per megawatt, do check your calculations again because you are likely to have made a mistake and this is for steam power plant with the typical parameter range of parameters if you use today. Remember that it is of the order of 1 kg per second per megawatt but it is an order that means around 1 is okay but if you go far beyond 1 say something like 10 or far below 1 something like 0.1 that quite often means that you have made a mistake. So do check your calculations again. Of course you can have for example a power plant based on low temperature waste heat where perhaps you will use if at all you use steam condensing temperature will be 40 degree C boiling temperature may be just 120 or 130 degree C but that is an atypical set of parameters and in that case perhaps you will have a steam rate much higher than 1 kg per second per megawatt. So if you are very far away from this particular what they say ballpark value or typical value do check out and see to it that you have some reasons to be so off from this. Now when it comes to power plants quite often we define the heat rate. The heat rate is in principle defined as 1 over the thermal efficiency of a power plant. So that means this is q dot supplied divided by the power output. The unit should be well one dimensionless. However the typical units used are in principle the unit should be joules again joule but since this is used mainly for electrical power this will be say kilo joule or mega joule per kilo watt hour. Again that is because kilo watt hour is the typical unit of electricity transfer for electricity consumption. So this is the heat rate. Then just the way that the specific flow rate and the specific output are measures of compactness. For example specific flow rate compare two plants with the same W dot net. A plant with a lower m dot W f will have smaller component because smaller flow rate means smaller pipeline sizes, smaller pipeline diameter, smaller component sizes. So any plant with a specific flow rate or steam rate lower compared to some other plant is definitely a more compact plant. So that is why you will find large capacity plants tend to be more compact for their size because they will have a lower steam rate. Then the although these are applicable to general plant there are similarly just similar to heat rate for power plants again we have what is known as the specific fuel consumption SFC. Now we know that the energy which is supplied comes out of burning some fuel. If it is a power plant which causes fuel like petrol, diesel, coal, oil, natural gas, whatever. So the specific fuel consumption is based on the fuel flow rate and the specific fuel consumption is defined as the fuel flow rate per unit power output and the units of this would be kg per second per watt or in the commercial terms for electrical output of plant it could be kg per kilowatt hour of course this can also be replaced by kg per joule. So these are the various units which are used. Finally there are of course my list is not going to be exhaustive. So this particular parameter is defined for reciprocating engines and this is known as the mean effective pressure. For its definition it is good for us to look at the PV diagram of such a machine. We have already sketched a PV diagram earlier when we talked of the work ratio instead of going back to that. Let us say that the piston moves between two extremes say V1 and V2 reciprocates bar back and forth and let us say that the actual PV diagram is something like this. So we know that this area represents work done per cycle. We also know that the power output, this is the network done per cycle, notice that it is only the loop area. This will be equal to work done per cycle multiplied by let me say N dot cycle which is what is known as the cyclic rate. The number of cycles executed per unit time it can be cycles per second typically or hertz if you want to use the electrical electronic terminology. But you can relate it to the speed of the engine in terms of rpm and the number of strokes in a cycle. So this is the thing and now the idea of a mean effective pressure is this. Suppose I consider this work done per cycle to be made up of some uniform pressure acting on the piston as it moves from V1 to V2. So if I put this area equal to this area and force this to be a rectangle then this by definition is PME. This has the nomenclature PME or mean effective pressure MEP. So that means this has to be equal to PME into V2 minus V1 and this V2 minus V1 in IC engines terminology is known as Vs stroke volume. It is the volume set by the piston as it moves from one extreme position to the other extreme positions. So that means the work done per cycle is PME into Vs. So in terms of this the net power output turns out to be PME into Vs into n dot cycle. So this is the definition for PME. You can write PME to be the power output per cycle or the energy produced, the work done, net work delivered per cycle per unit stroke volume. That is what you can define it as. But remember that this tells you something more about the engine. First thing it tells you that if you want to increase the power of an engine you should either change the stroke volume of the engine. That means make it a bigger engine, larger volume enclosed. So this is the size relation proportional to the size of the engine or you can execute more number of cycles in an engine, more number of cycles in a given time. So that means run the engine at a higher speed. Difficult to increase Vs because that is a basic design parameter. Maybe it is difficult to increase the speed beyond a certain limit because there are mechanisms involved. There are flow rates, lubrication rates and heat transfer rates involved. So given n dot cycle, given Vs, if you want to improve the power output you have to improve the mean effective pressure. That means you have to improve the area of this enclosed diagram. So that is the significance of the mean effective pressure. At this time you should tell them that for a typical IC engine it is possible if you instrument it properly to measure and plot such diagrams. Earlier we used to have mechanical devices, now we have simply electronic sensors and the diagram is shown on a computer screen. Such a diagram when plotted is known as an indicator diagram and this power which is calculated using this diagram is quite often known as indicated power. So we can call it IP or a better thermodynamic nomenclature is W dot net I indicating that it has come out of the indicator diagram. So this is the mean effective pressure. If you want at this stage you can introduce them to indicated power, introduce them to the brake power and if you do that you should notice that it is important for us that the indicated power is P mean effective multiplied by Vs multiplied by n dot cycle where remember that this comes out of geometry of the engine, this comes out of the operation of the engine whereas this is what is measured. So this is computed. We cannot directly measure it. A similar equation can be written down for the brake power. This will be PMe based on the brake power. Multiply it by Vs multiplied by n dot cycle and again Vs comes out of geometry, n cycle comes out of operation but the further things are different. It will be the brake power which will be measured using an appropriate dynamometer and the PMe based on the brake power will be computed. At this stage you can if you want introduce them to the mechanical efficiency of such reciprocating engines for that matter for any engine. This will be W dot net delivered as measured. This is the final delivery out of the engine divided by W dot net output as we can say developed as indicated by thermodynamic analysis, indicated diagram. And in terms of these two for reciprocating engines you can show that it is a ratio of PMe based on the brake power or the PMe as indicated or as determined or as measured using the indicated diagram. You may think that many of these are not properly definable because we have not described different engines and different cycles and I have said earlier that look that is a dilemma we have to face because either we define these things in general and then go to cycles or go on defining them as we explain the cycles. Now let us spend some time on looking at the thermodynamics. You know a cycle will be made up of a number of processes and we have a choice of these processes as constant volume, constant pressure and adiabatic. These are the three typical processes which we will have. Once in a while we will have an isenthalpic process or something slightly different but these are the three typical processes we have. We also have a choice of implementing these processes in an open system or in a closed system. Cloth system would be implementable in say a cylinder piston arrangement where the valves are closed. Open system would be implemented either in a cylinder piston arrangement with a valve open, at least one valve open or in other direct open systems like ducted open systems like a turbine, compressor, combustion chamber, pipe, seat exchangers, pumps, blowers etc. Again we have a choice of the working fluid which could be a gas or a vapor and in vapor apart from heating and cooling we also have evaporation and condensation as a choice. And let us see the result of this or the import of this. If you look at the processes let say constant volume process. This can be implemented in a closed system with valves closed and piston at one place. If I fix the piston in one place and keep the valves closed then it can be implemented in a closed system. But then all that we can do is either heating or cooling and unless it happens to be an internal combustion process, external heating cooling will always be external. It is likely to be a very slow process because heat will have to be transferred across the valve. So this is not a very common process unless we have an internal heat generation in many of the engines. In an open system it is very very difficult to execute a constant volume process because in a flow system keeping the volume constant or the density constant will be a very very difficult task. And pressure process is difficult in a closed system but can be implemented in an open system because we have the model of flow in a pipe. Arrange the velocity to be small so that the pressure drop is small. In which case we essentially have an isobaric process and if you heat the fluid during that process or cool the fluid during that process that is if it is a tube in a heat exchanger or a heater or a cooler or a condenser. We have a good practical model of an almost isobaric process and that is why this we will find is a very common process which we will come across. Another advantage of this constant pressure process is we should realize that a constant pressure process implemented with a vapor in which we have a process of boiling or evaporation or condensation with delta P small essentially implies that we have a constant temperature heating or cooling. And this is important because we have our background in the second law of thermodynamics saying that you know the Carnot cycle our ideal one with the best efficiency is something with the heating and cooling under isothermal condition. And this is a situation where without significantly worrying about the technology of simultaneous expansion etc expansion and heating we can have a flow system which is getting heated or cooled. So it is and if we arrange the flow situation in such a way that the pressure is small then in principle we have a isobaric system and in that if the flowing fluid is a vapor which is changing phase that means if during heating it is evaporating or cooling it is heating condense then this constant pressure process or almost constant pressure process means an almost isothermal process which is a very attractive proposition. And you will find that this also is reasonably common in practice. So indirectly we know how to under certain conditions execute an almost constant temperature process. Then comes the next one is the adiabatic process. Well adiabatic process only means the heat transfer should be small or heat transfer should be 0 or at least negligible and this can be reasonably well executed in either cloth systems by executing a process reasonably fast in a reasonably insulated cylinder. So you know a crude adiabatic process we execute with our hand pump by which we use to pressurize the tubes in our cycle, cycle tires similarly the compression of the charge in a petrol engine in a diesel engine is a reasonably adiabatic or approximately adiabatic situation. So is the expansion in a petrol engine and part of the expansion in a diesel engine. In open systems compressors and turbines are good candidates of equipment which executes adiabatic processes. After having done this let us look at the various equipment which we have. Let us say that what are the standard equipment which we have so that we can combine equipment and processes and create cycles. First we have the cylinder piston arrangement. In this constant v will be difficult, constant v possible but with internal combustion definitely possible. Others it is possible but may become very slow. Adiabatic yes possible, approximately at least to we have something called a turbine in which we have adiabatic expansion possible. Then we have a compressor open systems. This is the only closed system everything else beyond this is open system. Compressor well we have near adiabatic then we have a turbine a heater or a boiler we have constant pressure heating process and if it is a boiler we have constant pressure and temperature. And five if we have a cooler or a condenser we will again have a constant pressure heat exchange process and if it is a condenser we will have a constant pressure and temperature. And of course then we have some other pieces of equipment which we will need as needed. For example a throttling device may be used in a vapor compression refrigeration plant. Details we will explain when we come to that. Now notice that in any cycle let us say first look at the cycle for engines. We have seen Carnot which is TSTS and we have seen the purported advantages and the obvious disadvantages. We have also looked at Ericsson TPTP and Sterling TPTP. With a heat exchange Sterling which is TV TV again with a heat exchange and we have seen the advantages and disadvantages. But the real engines will generally be not of anything of this type. And the major reason for that is the constant temperature heat addition and constant temperature heat rejection apart from some other characteristics. So what we do is the real engines where begin with gas engines and in which if you see the compare either Ericsson or Sterling with the Carnot cycle you will notice that the isothermal heat addition and heat rejection are common to all three. What we have played around with is replacing the isentropic compression and expansion processes by either isobaric processes or constant volume processes. Why not replace give up the idea of isothermal absorption and rejection and replace it by either constant volume heat absorption and rejection or constant pressure heat absorption or rejection. So first one is we try to replace TSTS which is Carnot by TSTS that is heat absorption at constant pressure, expansion as in Carnot isentropically, heat rejection at constant pressure and compression like Carnot isentropically. I am told I am back on stream. So we were discussing replacement of the TSTS which is Carnot by PSPS. And this thing leads to the basic cycle which is the Brayton cycle assuming that the working fluid is a gas. And here we have heating at constant pressure which is done in a heater the working fluid flows through a pipe and externally supplied with heat we are just getting the idea. Similarly the cooling can be done by immersing the coil or exposing it to the environment so that the heat rejection takes place. And in between you have the isentropic expansion process and the isentropic compression process which is executed in two components the turbine and the compressor. Turbine, compressor, heater the turbine produces w dot t the compressor consumes compressor power w dot c and the net power is the power output of the turbine minus the power consumed by the compressor. If I show these four states like this fluid goes from state 1 to state 2 higher pressure, pressurized by the compressor 2 to 3 isobaric heating 3 to 4 adiabatic expansion. In the ideal case it could be reversible so isentropic expansion and then cooling completing the cycle. The if you neglect the pressure draw in the heater and the cooler on a T s diagram and on a P s diagram the cycle will look like this. Notice that first we should show the P s diagram because 2 3 is an isobaric process 4 1 is also an isobaric process. So it works between an upper pressure and a lower pressure P h and P l. You have 1 to 2 assuming reversible isentropic reversible adiabatic that is isentropic expansion this is the sorry isentropic compression. So this is the compression process followed by 2 to 3 isobaric heating process followed by 3 to 4 the expansion in the turbine adiabatic assumed reversible in the ideal case so isentropic and finally 4 to 1 isobaric cooling.  You can at this stage analyze the cycle and do a standard analysis of the gas turbine cycle sorry of the Brayton cycle or maybe I should use another page for this. Now the Brayton cycle which we have seen on the T s and P s diagrams uses a gas and because it uses a gas we can do what is known as a standard analysis. The standard analysis assumes the following it assumes that the gas is ideal with constant C p C v it assumes that all processes are ideal that means the gas is ideal. Wherever an isobaric process is present data p is 0 and it also assumes that whenever there is an adiabatic process it is also assumed to be reversible. So it can be replaced by an adiabatic reversible or isentropic process and because of this assumption this becomes a rectangle on the P s diagram and has such a nice shape on the T s diagram. But because of this under these assumptions if we do a standard analysis we can show that eta standard of a Brayton cycle standard means under the standard analysis assumptions will be 1 minus 1 over r p raise to gamma minus 1 by gamma where gamma is the ratio of specific heats of an ideal gas assumed constant and r p is what is known as the pressure ratio sometimes called the compression ratio for such a Brayton cycle and this is P h divided by P l and things to notice immediately are two parameters number one higher the r p better is the efficiency and number two higher is the gamma better is the efficiency and you can set up illustrative examples saying for the same r p if you use air which has a gamma which can be approximated well by the value 1.4 you will get one efficiency but instead of air if you use helium which has a gamma of 1.67 you will have a higher efficiency given the same pressure ratio r p. Now at this stage or at some other stage you can show them the actual Brayton cycle what we have seen so far is the ideal Brayton cycle remember that the ideal Brayton cycle was considered a closed cycle that is why we could consider helium as a working fluid but the disadvantage of this is that we have to supply heat from outside and we have to reject heat to the outside in a heat exchanger and the heat exchanger would become large bulky components not only costly prone to the technical problem but there will be significant heat transfer across there will be heat transfer large amount of heat transfer across temperature significant temperature differences which is not thermodynamic. The actual Brayton cycle compared to ideal Brayton cycle is an open cycle there is no cooler so one less equipment two it is an internal combustion cycle so air and fuel interact and burn react with each other inside the plant so the actual Brayton cycle plant will look like this you will have a compressor will take in air from the ambient so this arrow comes from nature the compressor compresses it to some pressure and then instead of a heater you have a combustion chamber in which fuel is supplied it is sprayed appropriate flow situation is created to have a stable flame so what goes in is air so at two also we have air but what comes out of the combustion chamber is gas that is air plus remaining air plus the products of combustion. This goes through the turbine so this turbine is given the name gas turbine and this cycle or this plant is known as the gas turbine plant or gas turbine power plant at four the gas is exhausted and the job of converting that gas back into the input air is left to nature so this is the actual implementation of a Brayton cycle plant you can sketch it on a TS diagram and if you want you can take care of some component efficiency for example you can determine what happens if the turbine does not have an isentropic efficiency of one but has a finite isentropic efficiency. Similarly you can determine what happens if the compressor has an isentropic efficiency less than one those details are there I think in some of the problems in the cycle exercise sheets but now let us look at what happens if we have Brayton cycle with vapor with a vapor as a working fluid let us go back to this PS diagram this PS diagram is good when the Brayton cycle is a gas but because the two processes are isobaric heating and cooling 2 to 3 for heating and 4 to 1 for cooling and the two processes adiabatic 3 to 4 and 1 to 2 are adiabatic or in ideal case adiabatic reversible processes this is also true this diagram when the thing is a vapor working fluid is a vapor the TS diagram is good only when the working fluid is a gas it is not correct when the working fluid is a vapor. Now let us see what would happen if the working fluid is a vapor on the TS diagram now we have to first sketch the vapor dome and the two isobars now this is the critical point the two isobars let them be shown like this this is the low pressure isobar PL and although I am not able to show it correctly here let me show the high pressure isobar now we have a choice of putting our two isentropic processes one of expansion and one of compression anywhere between the isobars we can start from here and go down we can start from here go down here can start from here go down here and start from here go down here for the expansion and we can have the compression either here or here or here or here if you notice that if I put the compression here and the expansion here creating this rectangle purely in the two phase zone as the rectangle representing my vapor Brayton cycle then you will realize that we have created an essentially Carnot cycle because although this process is an isobaric process and this process is also an isobaric process because during this process the pressure drop is neglected either boiling or condensation is occurring the isobaric process is also an isothermal process so inadvertently or by some trick we have created we have implemented a PSPS cycle but ended up being executing indirectly a TSTS cycle which is a very attractive thing now we realize that I can have the expansion process beginning from dry saturated steam and the compression process ending at saturated liquid we had I am just showing the earlier processes here I can go as far as this for my expansion process and as far as this for my compression process so my Carnot cycle the largest Carnot cycle which I can fit between a pH and PL is at shown here so I will show it as 1, 2, 3, 4 now when we implement this in actual practice what happens is we have no issue with the processes 2 to 3, 3 to 4 and 4 to 1 we can have a boiler which does this exactly for us 2 to 3 a turbine which does 3 to 4 and a condenser which does part of the condensation which is from 4 to 1 however when it comes to 1 to 2 what we are doing is we are compressing a two phase mixture of steam and air and after compression it is dry saturated sorry it is saturated liquid at a higher pressure. If you take some typical values for pH and PL we will find out that the compressor or the pump which we will be using we will need to tackle a extremely large reduction in volume of the working flow because one will have a reasonable amount of vapor in it which has a very large volume whereas at 2 everything is in a liquid state so the volume occupied is small. Such a pump or such a compressor is very difficult to design and even if it is designed we will have a very poor life in actual operation and hence what is done in practice is this process is removed instead of this 1 to 2 we modify the pump and when it comes to condensing the vapor we go right up to the saturated liquid state at the lower pressure use a pump to pump it to the higher pressure and then complete the cycle. Now because of this notice that the isobaric heating process is not fully isothermal it is isothermal from 2 to say 3 prime and then 3 prime to 3 it is isobaric 2 to 3 prime to 3 but it is isothermal only from 3 prime to 3 this. This earlier part 2 to 3 prime is heating at constant pressure of sub cooled water which will not be an isobaric process. If you look at the cycle this cycle is known as the Rankine cycle but it is nothing but the Brayton cycle with a vapor as a working fluid with the condition that at the end of the cooling process which is a condensation process we bring the liquid to the saturated liquid state we bring the condensate to the saturated liquid state then pump it up to the boiler pressure 2 to 3 prime is isobaric heating of sub cooled liquid 3 prime is saturated liquid 3 to 3 3 prime to 3 is evaporation. Now it turns out that since the critical point is only something like 375 degree C we still have the capacity to increase the temperature of the steam further along this line and it is very common for us to replace this process by a process at a higher entropy so you keep on heating it till you go to state 3 which is in the superheated zone making it still further away from the corresponding Carnot cycle expand it in a turbine. So quite often the Rankine cycle and this is a very common occurrence now looks like this so the isobaric heating process is partly 2 to 3 prime which is sub cooled heating up to saturated liquid state 3 prime to 3 double prime which is isobaric and hence almost isothermal heating to dry saturated steam and then 3 prime 3 double prime to 3 which is some amount of superheating so 3 is superheated steam. The advantage of superheating we need not emphasize this in a course in thermodynamics is to even show to them that apart from improving the thermal parameters like efficiency and steam rate it also increases the dryness fraction at the exit of the turbine which is good for the turbine from a technical point of view. The schematic for the block diagram for the Rankine cycle plant looks like this we have a boiler, we have a turbine, we have a condenser and we have a pump. State 1 is exit of condenser 1 to 2 is the pumping process the pump water goes into the boiler goes from 2 to 3 prime 3 to 3 double prime and then to 3 which is usually superheated where it is ducted to the turbine in the turbine it expands to 4 at which stage it goes into the condenser comes out as a condensate as 1. The energy interactions are heat supplied by combustion of fuel in the boiler or say Q dot B heat rejected in the condenser where cooling water is made available to extract the heat that will be Q dot C turbine power output W dot T pump power input W dot T. We can use our first law tricks to analyze all these and relate it to M dot which is the water or steam circulation rate and you will get W dot net equal to W dot T minus W dot C which should also be equal to W dot T minus Q dot C the efficiency would be cycle efficiency would be W dot net divided by Q dot B using the second form you can write it if you feel like as 1 minus Q dot C by Q dot B the work ratio if you want will be W dot C divided by W dot E and the specific output can be written down in terms of W M dot and W dot net. It is almost 1245 not only I am a bit exhausted by Q for 3 and half hours but I think up here now.


My name is Thomas Althaus. I'm a D-FIL candidate in Oxford University, but based in South East Asia. I've been mainly involved in a clinical trial based in Myanmar and Northern Thailand with the objective to guide antibiotic prescription in primary care. Prescription of antibiotics in South East Asia is is major. We consider this region the AP Center of Antibiotic Prescription in the world. And the consequence, the immediate consequence of this massive antibiotic prescription is the mutation of bacteria to resistant strain. And primary care represents the major source of this antibiotic prescription. And this is the reason why I was so interested in targeting primary care because it reaches a large amount of the community. And at the same time, there is no tool available to really identify patients in needs of antibiotics. You have to understand that a typical health worker in a primary care isn't a doctor. It's a nurse. We've a limited clinical training. We've probably more than 100 patients a day. So health workers are facing a difficult, challenging question. And they have no tool to really answer this question. Fever can be caused by different pathogens. When a fever is caused by a virus, it doesn't need an antibiotic to be treated. We call this treatment symptomatic. It means that you only need, for example, some paracetamol to control the fever and the body will handle the viral infection itself. On the opposite way, we also need to help them identifying those who have an infection that needs, that requires an antibiotic. Bacterial infection, for example, you can die from a bacterial infection if you do not have an antibiotic. The CRP rapid test is very simple per se. It's a test that tells you when to give an antibiotic or not. You can however imagine that patients might not be willing to comply to this test if they do not understand what it is. As an example, extremely surprised to see a patient asking for an antibiotic despite a negative test because it took him a day to reach the facility and in terms of cost of transportation that was also very high for him. So you cannot just simply expect people to comply to your prescription without explaining the test. What are the benefits and advantages of what we're doing and this implies to know and to understand our population. It's not about saying that antibiotics is bad. Antibiotics is extremely useful but it's precious and it's a scarce resource we need to control. So it's not a race to find new drugs. It's a race to slow down the pace of antibiotic prescription. Diagnosing malaria is extremely simple and cheap. We have rapid tests available everywhere for almost nothing and because of those tests being more and more negative because the disease is getting eliminated, we are now facing this new challenge that is once my malaria test is negative. What do I do with my patients? How can I manage a patient without malaria with a fever? How can I really figure out without having any access to laboratory if this patient needs an antibiotic or not? We also know that the most effective and funded research project trying to screen and to look for every pathogen do not discover more than let's say 50% of the causes of fever which means that even though you use your best diagnostic tool to investigate those causes of fever in 50% of the cases you will never know what they have. We know that antibiotic prescription in primary care is around 70% in Asia on average and bacterial infections should represent around 20%. So there's a huge potential impact to have on health at this level of care and with such impact for such small investments we could also allocate a bit of it to these non-specific illness tests that could save a lot of lives and at the same time immediately impact antibiotic prescription at point of care.


You don't think about the possibility of playing for such a long time for one franchise. It's just so rare in sports. I set up the game and I think it never gets old to say, but I was in a room for three years with a better player than me for the first time in my career. And I got to see what greatness look like up close. I got to see the difference between his game and my game and how I needed to improve to get better. I got to see a leadership style and take some things from it that I liked and make notes about things I want to do slightly different, which you should, you know, whoever follows me, that's the advice I would tell them. I'm fortunate to be able to play behind Brett for three years. I learned a lot and incredible lessons and and some great fundamentals working behind him, working with Tom Clements and Mike McCarthy and Joe Philvin. And that was meaningful, meaningful time for me. It gave me a deeper love for him and appreciation for him.


Everybody's farmer ready? Cool. Courtsies are good. RM slightly high. Got the old sign off by the Stumbia, so maybe get one. Here's your weather. Weather essentially said, baller day to fly, so clear everywhere. Ball it, so you guys have a clear step. It was good. Crush it on.


Hello and welcome to the session. Let us discuss the following problem today. If fx is equal to 4x plus 3 upon 6x minus 4, x not equal to 2 upon 3. Show that f for fx is equal to x for all x not equal to 2 upon 3. What is the inverse of f? Let us start with the solution now. We are given fx is equal to 4x plus 3 upon 6x minus 4. We have to find f for fx. We know f for fx is equal to f of fx.   This is further equal to x. I have 34 and 34 will get cancelled and we get f for fx equal to x. Every x not equal to 2 upon 3. Now let us understand how do we find the inverse? Let us consider a function f from x to y and g is any function from y to x. If g of f is an identity function on x and f for g is an identity function on y then g is inverse of f. We have to find the inverse of f itself and f for fx is equal to x that is the identity function on x. So the inverse of f is f itself. So we can write inverse of f is because f for fx is equal to identity function on x. So this is our required answer. This completes the session. Hope you understood the session. Goodbye.


In two dimensions, an equation in x and y defines a one-dimensional curve. For example, 3x plus 5y equals 15, well, that's a straight line. x squared plus y squared equals 16, well, that's a circle. x cubed plus y cubed equals x squared plus y squared to the fourth times. xy is a curve of some sort. Now, if we go up to three dimensions, an equation in x, y, and z is going to define a two-dimensional surface.  And so, given an equation, we should be able to use level curves to sketch the surface. And we should also be able to go backwards. Given a surface, we should be able to write an equation. So, for example, let's say we want to find the equation of the sphere of radius 4 centered at the origin. So, here it's helpful to remember that a sphere consists of all points a specific distance from its center. Now, the sphere is centered at the origin, so we want the distance from any point x, y, z on the sphere to the origin to be equal to 4. Now, the Pythagorean theorem holds in three dimensions, so we could use the distance formula. The distance between the two points is the square root of the sum of the squares of the differences. And we want that equal to 4. And we can simplify this. And we have a square root, and we can square both sides to eliminate the square root to get a nicer looking equation. Well, what if we try something like a plane? Well, we've already found some equations for planes. z equals 0 is the equation for the x, y plane. y equals 0 is the equation for the x, z plane. And x equals 0 is the equation for the y, z plane. And we've also graphed some of these. z equals 5 is a plane. 3x plus 2y equals 12 is a plane. 2x plus 3y plus 5z equals 30. Well, when we graphed it, we saw that it looks something like this, which also suggests that it's a plane. And this suggests the following. The graph of ax plus by plus cz equals d is a plane. So let's find some equations. Well, you know from geometry that we need three points to specify a plane. So let's find the equation of the plane through these three points. So the equation of a plane is ax plus by plus cz equals d. Now, the important thing to remember is we don't know a, b, c or d. So these are our variables. And we can solve for them by setting up a system of equations. So first, remember, if a point is on a surface, it makes the equation for the surface true. And since 2, 1, 5 is on the plane, then if I let x equals 2, y equals 1, z equals 5, I should get a true statement. So substituting in those values for x, y, and z. And so we know that 2a plus b plus 5c must equal d. Well, that worked well enough. Let's take a look at our second point. We know that 1, 4, negative 3 is on the plane. And so that gives us the equation a plus 4b minus 3c equal to d. And again, 1, 2, negative 4 is on the plane. And this will give us an equation. And now we have a system of equations. And we can solve this system of equations. First, we'll reorder the equations. We don't really need to. It's just convenient if we do so. And as we'll see in a moment, it turns out to be very convenient if our leading variable has coefficient 1. So we'll make one of these two equations our first equation. And then the others can be in any order that we want. So let's try to solve the system of equations. And so notice that the first two equations have the same coefficient on a. So if we subtract the first from the second, we'll eliminate that a. And we don't need that second equation anymore. The third equation has coefficient of a twice the first. So if we subtract twice the first equation from the third, we can eliminate the variable a. And that gives us a new third equation. And again, we no longer really need that original third equation. Now, in our second and third equations, the coefficients of b are different. But if we multiply the second equation by 3 and the third equation by 2 and add, we eliminate b and we get a new third equation. Now, since we have three equations, but four unknowns, we can try to pick a value for the fourth unknown and compute the others. So notice that if we choose a value for d, then finding c will require dividing by 29 and finding b will require dividing by 2. So for convenience, we might choose d to equal 29 times 2. That way, when we have to divide by 29 or by 2, at least we're starting with something that can be divided by both and hopefully will avoid fractions. So if d is 29 times 2, then our last equation becomes and we solve. And so that gives us d equals 29 times 2, 58, and c equals negative 4. So now that I know c and d, we can substitute into our third equation and get and solve. And now that we know b, c, and d, we can substitute into our first equation and get our value for a, b, c, and d, and so our equation will be. And you might notice that each coefficient has a factor of 2, so we could reduce this equation, but we won't bother. What if we wanted to write the equation of a line? Well, let's make a few observations. Notice that in two dimensions, a one-dimensional straight line has an equation with two variables, the line 3x plus 5y equals 60. But a zero-dimensional point requires two equations, the point where x is equal to 5, y is equal to 8. And that's because the point is at the intersection of two lines. And in fact, if you think about it, x equals 5, y equals 8 are the equations for two lines. Now, when we go up to three dimensions, in three dimensions, a two-dimensional plane has an equation with three variables. But if we drop down and try to consider a one-dimensional line, this is going to require two equations.  And so if we want to talk about lines, we have to say the line at the intersection of two planes where we give the equations for the two planes. And what this means is if we wanted to find the equation for a line through two points, we'd need to find the equations for two planes. Now, we could do this, but maybe there's an easier way. And in order to get there, we'll need to introduce some more ideas first. So we'll take a look at those next.


This episode brought to you by healthwithdronetech.com Ageless multi-collagen provides you with the five types of collagen your body needs and one complete protein as an added bonus Ageless multi-collagen is non-gmo and there's zero artificial flavors colors preservatives sweeteners gluten or antibiotics get ageless multi-collagen for 51% off plus free shipping by going to www.healthwithdronetech.com or by clicking the link in the video description box below But so here's the situation though. We have there are millions of Americans Almost all white almost all Republicans who somehow need to be deprogrammed there It's as if they go they they're members of a cult the Trump is called And and we have to be deprogrammed. Do you have any idea how we? How we start that process? What is it with this group of people the media and their seemingly inherent desire to just wipe out all political opposition? I mean really they're always having these fantasies about their political opposition dying off as if everybody who opposes them is just some old White person or they want us all deported or cleansed like this ABC News reporters Suggested needs to happen Trump will be an ex-president in 13 days The fact is that getting rid of Trump is the easy part Cleansing the movement he commands is going to be something else Welcome back everyone as of right now the Democrat Party and their media have basically unchecked power and Are currently using it to wipe out the ability of their political opposition to have a voice or even participate in the system They're basically creating two Americas with a two-tier system of standards rights and laws They can protest you can't they can riot you can't not that I'm in any way endorsing riots I've consistently been against them no matter who they're coming from But let's be honest here the vast vast vast majority of violence and writing that we see in this country comes from the left They can have their own social media platforms Places where they can meet up discuss topics and organized Places where their nebulous standards are used one way against those they disagree with while you're not even allowed to have your Own platform their elections are legitimate while any time you win the country is put through four to eight years of Resistance from the Democrats in their media you get the idea What's becoming increasingly disturbing is the language that's being used by powerful leftist in the media about their fellow Americans who just happened to disagree with them politically not only do you have them talking about Reprogramming and cleansing those that are blocking their total power But you even have the principal counsel for PBS saying that they want to do some pretty heinous things the Trump supporters now before I go Further just a quick note I just want to say this video does seem to be edited as videos tend to be But I haven't seen the full version yet and I can't say for certain that this video hasn't been manipulated in some way The White House Kids who are growing up knowing nothing Kids Great They're nice. They have Sesame Street characters in the classroom And they watch PBS I Gotta tell you I don't have a hard time believing that this video is real Just based on the four-year hatefest that our media has subjected this country to this does remind me of another veritas video That I don't think was ever discredited showing a Bernie Sanders field organizer who is now the second staffer to be caught praising Communist totalitarianism he says he wants to kill people in violent revolution and put counter revolutionaries in prison Seeing as how we're against communism. I guess that would make us the counter revolutionaries I'll sort of get armed. I want to swing on a shoe and go train. I'm ready He attains the rip is when we send all the Republicans You imagine mixed McConnell, oh god, he wouldn't survive a tiny grail Like you guys are founded as re-education, right? The first school life that was open. Have you heard about the Bellamore Canal? I How about the California regional director of the DNC David Atkins Who suggested quote seriously? How do you D program 75 million people? Where do you start Fox Facebook? We have to start thinking in terms of post-World War two Germany or Japan. This is not your standard partisan policy disagreement This is a conspiracy theory fooled belligerent death cult against reality and basic decency No, this is literally the Democrats in their media labeling anything that stands in opposition to them a conspiracy theory folks These people want you to think that Trump came out of nowhere and then suddenly everyone distrusted the media and started thinking that Democrats were Communists, it's actually nothing new for as long as I've been in the politics since the W. Bush administration We've been having the exact same arguments The only thing that changed that the right started to get a better foothold against the left in these culture wars And then as soon as we started blowing up on social media, they had to put it into it now They think they're gonna fix that by any means necessary and as you can see here Their ability to rationalize it's astounding with mainstream discussions of re-educating 75 million Americans for having a difference of opinion. That's it for this one Please make sure to hit that like button and share as nobody even knows I exist anymore I'm not sure how long I'm gonna be around here given the current insane climate So make sure to follow me on my main website on gab on bit shoot on odyssey on rubble Dlive and cloud hub. They're not gonna get rid of me that easily if you want to support this channel or follow me on any of Those other platforms you can find all those links in the description or pin comment. 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