Felly, i ôl dim ond idde yn gweithio gynnig peth i'r rhaid i gondol tîm i'r ffawr i bryg o'r anghyddant o'r cyrgylchau cyffredinol o'r argyfnodol Beili Gwyn. The cabinet secretary will take questions at the end of his statement and so there should be no interventions or interruptions. I would ask members who wish to ask a question at the end of the statement to press their request to speak buttons whenever they wish. Rwyf ni'n edrych i gychwyn i'r sôn a'r byw,ㄱrgo fydd y celf. Mi'n rhan o'r byw ond gan yr ystafell e'n fwyaf i Gwyn-Gwyn. Rwyf ni'n rhan o'r byw ac yn cyflawni'r hyn,yn i ddysgon ni'n erbyn ni'r reilio ac arnyn ni yw ddignitiadau i gyd-di, fel艙d i Gwyn-Gwyn, dwy gyrfaenol.  Felly, rydа i ddyfod mwynahydd i Llywodraethau Aparwyr i Gwydiannau Gwydiannau i Llywodraeth a Ch another sefydlu i Gwydiannau Gweithgraf i Gwydiannau Gwydiannau a Rhaidau. Aberdeen City Council has responded effectively and swiftly, commissioning a review into the issues that gave rise to this incident and also in taking forward recommendations from that independent review. As that report makes clear, Bailey's death at school, whilst very rare, was a shocking and tragic incident, it was an unplanned and spontaneous conflict from which we must learn to minimise the risk of it happening to other children and young people. The resilience of schools in addressing the threat posed by weapons and violence is key and I now wish to set out the Scottish Government's response to recommendations 11 and 12 from the review. Both recommendations have been considered carefully and ministers have been mindful of the need for a response that addresses what happened to Bailey and his family, but also would impact on all of Scotland's teachers, children and young people. I want to be absolutely clear that the safety of our children and young people at school is paramount. The possession of weapons in schools is absolutely unacceptable, as are threats and violence towards pupils and staff. I will respond to recommendation 12 first on behalf of the Government, with the Cabinet Secretary for Justice's full support. That recommendation asked us to explore further legislative controls that can be brought to bear on the purchase of weapons online. I can confirm that we are doing so. The Scottish Government has the ability and has legislated to provide additional controls on the possession and sale of knives and offensive weapons. It is already illegal in Scotland to possess a knife in a public place without reasonable excuse. In March 2016, we increased the maximum penalties for offences of handling offensive weapons and knife possession from four to five years. Schools are included within the enforcement of those laws. It is also illegal to sell knives or similar products with blades or points to anyone under 18 except the sale of knives for domestic use being permitted to those over the age of 16. A local authority licence is required to supply knives to the general public, not for domestic purposes. The licensing scheme, which has no equivalent in England and Wales, aims to ensure that those who sell such knives do so responsibly. Anybody selling a knife to somebody below the minimum legal age risks a fine of up to £5,000. However, we know that in the case of Bailey's death and indeed in the sale and purchase of many weapons these days, all those legal requirements and processes were circumvented through internet activity. We can act to change the law in Scotland on the purchase of knives, but as the purchase and delivery of knives crosses the borders of all UK countries, it is clear that the impact of a change only in Scotland would be limited. The most effective way to ensure more robust controls are in place would be through UK-wide action. Accordingly, the Cabinet Secretary for Justice has written to Sarah Newton, the UK Government's Minister for Vulnerability, Safeguarding and Countering Extremism, to seek agreement to a UK-wide approach to address concerns about the online sale and delivery of knives. I hope that the Parliament will agree with and support that approach, and the Cabinet Secretary for Justice will update Parliament once our response is received. Turning to recommendation 11, I want to reiterate that our approach to education places the health and wellbeing of children and young people at the heart of our curriculum. This begins with a whole school ethos, which emphasises the importance that positive relationships play in maintaining and building resilience, providing a protective environment, supporting vulnerability and addressing adversity. It is worth noting that the OECD review of Scottish education published in December 2015 identified that Scottish students are resilience as one of its positive developments in terms of those performing in the top quarter of international achievement. While our approach, backed by tough enforcement when necessary, is working, we must be prepared to do more. In the rare instance that violence does occur, we take those cases extremely seriously. The safety and wellbeing of everyone in our schools is our top priority, and where concerns are identified, robust action will be taken. I have sought stakeholders' views to inform our response to recommendation 11, and I can advise Parliament that I will include new guidance on violence and weapons in schools within our refreshed guidance on school exclusions. That will be published this spring. The new guidance will make clear that any incident must be monitored and recorded at a local level. Indeed, Aberdeen has undertaken exemplary work to put such processes in place already. Monitoring and recording will enable local authorities to review and to improve local policies. The Government's approach to the issue of violence has been consistent. By investing in early intervention, we can deter and divert people from harmful behaviours and can encourage positive relationships that are founded on respect, tolerance and inclusion. We have already invested significantly and will continue to do so in activity delivered in partnership with schools and a range of agencies and organisations. Those will continue to be freely available to all schools. Since 2007, we have invested more than £10.5 million in activity to reduce violence among young people, including more than £3 million for no knives, better lives and more than £7.6 million since 2008 in the national violence reduction unit, including delivery of the mentors in violence prevention programme. The programme seeks to give young people the skills and confidence to safely intervene and to speak up to protect themselves and their friends against violence and abuse. We are now accelerating the expansion of the MVP programme to reach an additional 30,000 young people in another 93 secondary schools across Scotland by March 2018. A total of 108 schools across 18 local authorities are currently engaging with the programme. Recommendation 11 also asked the Scottish Government to give consideration to amending the law in relation to searching pupils. I want to assure Parliament that I have given very careful consideration to this point and listened to advice from key stakeholders, especially teachers and their representatives. I can advise Parliament that I will not be taking that recommendation forward. Schools and local authorities in Scotland already have robust processes in place to address concerns about violence and weapons. Those take into account health and safety issues and are based on risk assessments that enable staff to deal appropriately with situations where a weapon is suspected. In such circumstances, teachers may ask to carry out a consensual search. Changing the law would confer statutory powers on teachers, allowing them to compel a young person to be searched. Currently, outside of the prison system, this power is held only by the police. We would therefore be placing teachers on the same footing as police officers if we were to change the law. That would radically change the teacher-pupil relationship, which is often fundamental to encouraging young people to change challenging behaviour and potentially damage the school ethos and commitment to positive relationships that currently exist in Scottish schools. Given the recent debate on stopping search powers for Police Scotland in relation to children and young people and the necessary safeguards that are now in place, it is important that we take that consistent position into our schools and communities. We will continue with the current approach while strengthening and clarifying the position in our refreshed guidance on school exclusions. The guidance will be clear that consensual searches can continue, but if a teacher is uncertain or a young person will not cooperate by showing their belongings, then the police must be called immediately. I want to reassure Parliament that the justice secretary and I have given careful consideration to those two recommendations. We have sought advice and listened to a range of views and experiences. We have taken seriously our responsibility to provide an adequate and appropriate response on behalf of Government to the changes that are recommended to us. Most important of all, throughout our deliberations, we have kept at the forefront of our minds that a young man lost his life in one of our schools. We recognised that it was incumbent on us to respond in a way that not only minimised the risk of that happening again, but also acknowledged all the circumstances that led to Bailey's death and the harm and hurt that caused to his family. Taking all of that into account, I believe that the response that I have set out to recommendations 11 and 12 in the independent review is the correct one. It is a response that demonstrates this Parliament and I believe that this Government and this Parliament's absolute commitment to ensuring the safety and wellbeing of all our children and young people in the schools of Scotland. The cabinet secretary will now take questions. We will have about 20 minutes for questions. If members wish to ask, please press your request-to-speak button now. Thank you, Presiding Officer. I thank the cabinet secretary most specifically for his very measured and sincere tone with which the Scottish Conservatives would like to associate themselves. I understand what the cabinet secretary has said in relation to the debate about whether or not teachers should have additional powers to search for weapons. Specifically, the point made in the advice that has been provided to the Scottish Government that those additional powers would place teachers on the same fruiting as police officers and that that, by definition, would change the pupil-teacher relationship. I can therefore accept the reasons provided by the Scottish Government as to why additional powers would not be acceptable and why they would be inconsistent with other policies, and I think that the decision will be warmly welcomed by teachers who I know had great concern about the possibility of extra powers. Notwithstanding that decision, the cabinet secretary makes clear that, in certain circumstances, as now, teachers may be permitted to carry out a consensual search and that further guidance on this will be issued in the spring. Could I ask him to make sure that, within that guidance, it is abundantly clear whose consent must be sought, whether, as well as the pupil, it would include the head teacher and the parent or guardian? Is there any discussion that is taking place with the GTC about teacher training on circumstances that, hopefully, will not be ever repeated again but nonetheless have a possibility of repetition in schools? May I draw Parliament to my register of interest as a member of the GTCS? First of all, can I welcome the response of the Conservative Party and the comments made by Liz Smith and indicate that the remarks that she has made about the difficulties and the dilemma of the judgment to be applied here are exactly the questions with which the Justice Secretary and I have wrestled in the course of the last few weeks? I want to make it absolutely clear that, when the Government is formulating the guidance that is issued to schools, we will do so in consultation with the long-established mechanisms for dialogue on the subject involving our discussions with the Scottish advisory group on relationships and behaviour in schools, which are obviously actively consulted as part of the process. That brings together the directors of education, the Convention of Scottish Local Authorities, the teaching unions and a variety of other stakeholders to make sure that that guidance can be very clear and clearly to be followed by members of staff in our schools. That brings me on to the question of equipping our teachers to ensure that they are able to handle such situations and, of course, as part of the approach to initial teacher education, but also in relation to the on-going teaching of members of staff and in the leadership positions within schools, it is important that teachers are equipped to handle the situations that they face. The emphasis that I have placed in my statement on the encouragement of positive behaviours is strategically for the Government a very important commitment. We think that that accords with the ethos that is prevalent in Scotland schools and that it is an ethos that must be encouraged and nurtured, but, equally, teachers must be able to be equipped to handle situations that we hope they do not have to face but, regrettably, in this circumstance some teachers have had to face. I give Liz Smith the assurance that we will take due account of the need to ensure that teachers are properly trained for all circumstances and that the guidance that we issue is sufficiently clear to address the issues that she has raised. Thank you, Presiding Officer, and I thank the cabinet secretary for an advice site of the statement. The death of Bailey Gwyn was tragic in our thoughts are with his family and friends. The decision to undertake a review and give all recommendations serious consideration is welcome and I appreciate the work that is already carried out by Aberdeen City Council. We support the Government's position on recommendation 11 and its analysis of this proposal. Principally our schools must be places for learning, parents must be confident that their children are safe and supported by teachers who have the relevant and proportionate powers and responsibilities to do so. We support all efforts to restrict the purchase of weapons online throughout the UK and support the approach that has been made to the UK Government. However, could the cabinet secretary say more about what changes could be introduced to the law in Scotland, while accepting that it would be limited, as would UK restrictions in the context of worldwide online sales? Finally, recommendation 20 of the report depends on the completion of the Scottish Government's child protection system review, which was due to report in December. We are still awaiting its publication. Can the cabinet secretary update Parliament on when that will be published? First of all, I welcome Clare Baker's commitment on behalf of the Labour Party and its understanding and support for the position that we take. The way that Clare Baker described our schools is exactly the way that I would describe them as well as places of learning of safety for our young people and environment of support by the teaching profession. That is the approach that we have to work to sustain in the context of those recommendations. In connection with the—I also welcome her support for the efforts that the justice secretary is taking forward in trying to get to a UK-wide position. We have come to the conclusion that that would be the most effective way of trying to take as much action as we could to address the deficiencies in online security around the purchasing of knives. We think that taking forward actions in Scotland, particularly on requiring more stringent measures on age verification, could be undertaken and could be applied, but the danger is that we would not be able to capture all of the potential sources of knives to be dispatched from online sales. Although, even within a United Kingdom measure, we have to accept that that might not be possible because of the nature of business traffic that is currently undertaken. We will explore with the UK Government how we can most effectively strengthen those provisions and, as I said in my statement, the justice secretary will report to Parliament on that question and ensure that the updates that we receive from our dialogue with the UK Government are reported to Parliament. On her final issue, the Government is still giving consideration to this issue and reports that we have published in due course. Jenny Gilruth, to be followed by Ross Thompson. I remind members that I am the parliamentary liaison officer to the cabinet secretary for education. To ask the cabinet secretary what the Government is doing to support schools in developing robust security and safety measures that ensure the wellbeing of all pupils and staff. Much of the activity relates to ensuring that we create a safe ethos within individual schools. The proactive measures that we take, such as through the work of the violence reduction unit, the mentoring programme and the communication of the no knives better lives campaign are all designed to create that positive environment in which young people can be supported effectively within our education system. I have set out in my statement today additional measures that we will put in place to strengthen the guidance so that there is more effective and assiduous reporting of incidents and, as a consequence, more effective and assiduous follow-up to those incidents to ensure that we learn lessons and are constantly trying to make our schools as safe as they possibly can be to protect the wellbeing of young people in our country. Ross Thompson, to be followed by Lewis MacDonald. Thank you, Presiding Officer. I declare an interest as a current and serving councillor on Aberdeen City Council. As a north-east member representing the community of cults, I know how deeply affected the school community and the wider community have been by the tragic death of Billy Gwyn over a year ago. It is welcome that, following the publication of Andrew Loh's report, Aberdeen City Council agreed to a new strategy aimed at preventing knives and other weapons from getting into schools and acted swiftly to implement the report's recommendations. Will the Scottish Government work to ensure that the exemplar work that is carried out by Aberdeen City Council in response will be rolled out by other councils in Scotland? I have already indicated my appreciation of the work that has been undertaken swiftly and timidly by Aberdeen City Council. The council has addressed the issues that have been raised in the independent report very swiftly and effectively. There is good learning to be undertaken here, and what has been developed by Aberdeen City Council is a practice that will be significantly influential in the guidance that the Government looks at in the further discussions that we have before its publication later this spring. As the parent of a former pupil at the cults academy, I thank the cabinet secretary for acknowledging the impact of Billy Gwyn's death on the school and the community and above all the family who have acted with the utmost dignity in the most difficult of circumstances. He has, I think, already confirmed the discussions that he has had with Aberdeen City Council. Will he confirm further that the model of joint working with the other public sector bodies involved in this case has a direct relevance to other such cases where different public authorities have a responsibility and those responsibilities intersect? If indeed the implementation of recommendations for Aberdeen City Council and the other public bodies is indeed well on course, will he tell us whether there is any further role that the Scottish Government expects to play in supporting the council on those matters, particularly in relation to the school and the family? First of all, I readily acknowledge the impact of this tragedy on the cults academy, on the community and most especially on Billy Gwyn's family. I think that the school community has handled what is an almost unimaginable circumstance with all of the dignity and the grace that one could ever hope to summon in such circumstances. Mr MacDonald is absolutely correct that none of those responses take place in isolation. They must be the product of good collaborative working between different public organisations. I can confirm having discussed those issues with the director of education at Aberdeen City Council just how appreciative she is of the work with Police Scotland and particularly the family liaison officers who undertake a most extraordinary task on our behalf in such very difficult circumstances to draw all of that together. It is important that that partnership remains in place. From my conversations with the director of education last week, I am absolutely sure that that is the case. The Scottish Government will work closely with Aberdeen City Council on taking forward any areas where we can be of assistance in advancing the agenda. Clearly, a lot of that joint working will be reflected in the guidance that the Government brings forward, and we have been given some significant assistance in developing our thinking on that by the actions of Aberdeen City Council on that issue. Fulton MacGregor, to be followed by Liam McArthur. Can the cabinet secretary provide more information about the No Knives Better Lives campaign and how it has impacted on the prevalence of knife crime in communities where the programme has been operated? The No Knives Better Lives campaign has been in existence since 2009 and it has been applied across 19 local authorities at the present moment. The consequence of that activity is to drive a significant reduction in the existence of knife crime, and it is particularly focused on changing behaviours among young people, particularly in the 11 to 18 age group. We have seen significant progress in that respect and we look to sustain that with the commitments that the Government makes today. Liam McArthur, to be followed by Julian Martin. I thank the cabinet secretary for early sight of his statement and associate myself with his remarks and the remarks of other colleagues about the dignity shown by Bailey Gwyn's family at what has been an exceptionally difficult time for them. In relation to the response to recommendation 11, I warmly welcome the approach that has been taken. I do not underestimate the balance of issues that had to be weighed up, but would he accept that going down the route of giving teachers statutory powers of search would have run the risk of undermining confidence, trust and relationships between staff and pupils that arguably could have made school environments more difficult in dealing with challenging behaviour? In relation to the online sales, is there any data or evidence that would show where online sales are coming from? One would imagine that it is probably international, but within the limits of the powers that he has, I would certainly support the approach that has been made to the United Kingdom Government. I thank Mr McArthur for his support and say that the judgment that I arrived at on recommendation 11 was essentially by wrestling with the dilemmas that he put forward there. My judgment was that we would have potentially recast fundamentally the relationship between teachers and pupils in a way that was completely at odds with the approach of our education system, but we would also have created an approach on search powers that would have been at odds with the prevailing direction of the very good work that the justice secretary has led and been informed by the working group that was led by John Scott, which has created the guidance that Parliament is currently scrutinising on the very question. We have taken a decision that is rooted in our views about how the education system should develop and our consistent approach on stop and search. In response to his question on online sales, it is a more difficult question for me to pin down, but, in my answer to Claire Baker, I accepted that there is a difficulty here because of the nature of the international market. We have accepted some part of that in the sense of trying to pursue a UK co-operation to try to cast the net as wide as we possibly can do, but I do not underestimate the fact that there could be dangers because of international activity. I think that it strengthens the argument for saying that retailers must be mindful of their conduct and their behaviour in relation to the sale of knives and the dispatch of knives, because we know from the report that was undertaken in this case that international activity was involved. There was a certain amount of approach that was taken to circumvent the controls that were put in place in relation to purchasing. We have to be mindful of the fact that not all of that activity can be policed by the Scottish Government, but I have tried to reassure Parliament about what we will do as much as we possibly can do to address that directly. Gillian Martin has been followed by Douglas Ross. Cabinet Secretary, I am sure that your statement will be welcomed by families and teachers in this school across Scotland, in particular the forthcoming guidance. Can the cabinet secretary advise what input teachers have had or will have into the drafting of this guidance? The guidance is developed in consort with the Scottish advisory group on relationships and behaviour in schools, which brings together the teaching unions, the Association of Directors of Education in Scotland and the Convention of Scottish Local Authorities. We will also be very mindful of other input that we take from key stakeholders such as the violence reduction unit working with the Government. We will make sure that the views and the input of teachers is at the core of designing the guidance that is rolled out to schools in Scotland. Douglas Ross will be followed by Colin Beattie. The cabinet secretary will be aware that, on page 27 of the independent review, there is a quote from the chief superintendent, which stated that, at the time of the review, the local force had been called out to six incidents at local schools, including three in Murray, at Bucky, Keith and Elgin. However, the statement also concluded that all incidents were reported by school staff to the police. Given that statement, recommendation three of the report still went on to say that Police Scotland shall be notified of each and every incident of weapons possession of which the school become aware. Was that recommendation included? There are still concerns that some incidents are not reported by school staff to the police. What more can be done by the Scottish Government to ensure that each and every incident is reported by school staff to the police? What I said in my statement is that the guidance will be clear that consensual searches can continue, but if a teacher is uncertain or a young person will not cooperate by showing their belongings, then the police must be called immediately. There is a very direct relationship between the importance of involving the police where there is a suspicion of a weapon being concealed that is not—over which there is not co-operation in undertaking that search. The guidance that the Government puts out will address those issues as part of creating an approach within our schools that tries to resolve issues, in the first case, by encouraging young people not to carry knives in the first place. All of that will be put into to promote that message, that strong contribution to the ethos of our schools. There will then be guidance about the ability of teachers to exercise control and to undertake consensual searches if a young person is prepared to co-operate, and if that is not successful, then obviously to involve the police as is the appropriate way for such matters to be handled consistent with the approach that I have set out today. Colin Beattie, to be followed by Daniel Johnson. I welcome the news that the Scottish ministers want to see the online purchase and receipt of offensive weapons outlawd. Can the cabinet secretary advise when he expects to receive a response from the UK minister and what action Scottish ministers will take if the UK Government refuses to act to address this issue? The cabinet secretary for justice wrote to the Home Office on 18 January, so we will wait some time before we hear a response. Obviously, we hope that we can encourage the UK Government to work with us on this question and to find practical ways of addressing it. If we are unsuccessful in that respect, then obviously I come back to my response that I gave to Clare Baker that the Government will do as much as it can within the confines of our powers and responsibilities to ensure that we can be as effective as possible in this respect. Daniel Johnson, to be followed by Clare Adamson. Amongst the recommendations that were made in the independent report to Aberdeen City Council were a recommendation 10 that they worked with the Benconciller Trust on violence reduction policies, and a recommendation 13 that they consult with respect me regarding anti-bullying policies. I note and welcome the comments that the cabinet secretary made around violence reduction measures in programmes, but following on and by extension from those recommendations, what contact has the Scottish Government had with those organisations and others around those matters and what revisions and improvements have been made to the violence reduction programmes in light of the tragic circumstances of Bailey Gwinn's death? The violence reduction programmes have been long-standing programmes, well-developed and I think very successful in reducing crimes of violence among young people, and particularly around knife crime and the statistics speak for themselves in that respect. So I think that the programmes have got a lot to assist us with in this respect. On Mr Johnson's two points about recommendations 10 and 13, in relation to respect me, obviously respect me act in taking forward work on the Government's behalf in working to tackle the issue of bullying among young people. As Mr Johnson will know, there is some guidance on bullying strategies that is currently being consulted upon. We are engaging with the Equal Opportunities Committee of Parliament on that question, and I hope to draw matters to a conclusion on that approach once I hear further from the Equal Opportunities Committee on their thinking. I can assure Mr Johnson that we work very closely with organisations. Such respect made focused on working to tackle bullying within our schools and to provide the most effective support to assist in that respect. I commend my condolences to the friends and family of Bailey Gwynne as well as my colleagues this afternoon. With reference to the answers that you have already given regarding the consultation with the teaching unions and with other stakeholders across Scotland and also the information about the consistency of recording the data, can I ask what steps you will take to ensure that there is a whole Scotland approach to that across the 32 local authorities and what support the unions would be able to offer in achieving that goal? The guidance that we put out will obviously apply across all local authority areas, and I have been greatly encouraged by the degree of common thinking and support that has been expressed by the Convention of Scottish Local Authorities for the approach that we are taking and the approach that has been supported by the Association of Directors of Education in Scotland. On that question, the existence of guidance that has an applicability right across the country and the willing participation of a range of different organisations will help us to take forward what is an important agenda to safeguard the wellbeing of young people in Scotland's schools. Thank you, cabinet secretary. Members, that concludes our statement. We will now move on to a debate on motion 3748 in the name of Fiona Hyslop, and we will just take a few seconds to change seats.


air enters the compressor of a regenerative stationary gas turbine engine steadily at 100 kPa, 27°C and 5 kM per second. The engine features a pressure ratio of 10, and is designed to induce a maximum temperature of 1400 Kelvin. Assuming the cold air standard and that the effectiveness of the regenerator is 80%, determine and complete the following and compare the results to Brayton cycle number one. First I want us to determine the specific network output of the cycle, and then the thermal efficiency of the engine. I will start with the diagram we had generated when we were talking about regeneration, and in order to complete the cycle here I will add in a cooling process between 6 and 1. That's the Q out box that we've been analyzing, which again doesn't exist. And then I recognize that my work in is only occurring in the compressor, my work out is only occurring in the turbine, my Q in is only occurring in the combustion chamber, and my Q out is only occurring in the Q out box. And to be consistent with the rest of the energy components I had written, I will write that as a specific Q out instead of a Q dot out. The regenerator itself is treated as being adiabatic, and all of the energy that is exchanged is exchanged internally. That means it doesn't cross my system boundary when I analyze the entire cycle. So when I'm calculating Q in, Q out, and Q out for the entire cycle, I don't include Q regen, because again it doesn't cross the boundary of the entire cycle. Anyway, I know that I'm going to have six state points here, so I will start generating a table. I know that I have an inlet temperature and pressure, that's state one's temperature and pressure, of 27 degrees Celsius and 100 kilopascals, and I know I have a pressure ratio of 10. And because all heat exchange processes in the Brayton cycle are assumed to be isobaric, that means that I actually know all of the rest of my pressures. Because P2 will be after the compression process, therefore it will be 10 times P1, and then P2, P3, and P4 are all the same pressure, and P5, P6, and P1 are all the same pressure. So this is a thousand, this is a thousand, this is a thousand, because 10 times 100. This is 100 and this is 100. Then I was told the maximum temperature was 1400 Kelvin, and I know that the maximum temperature is likely to occur after the heat addition process, so we will treat it as being T4. Remember it's always hottest after the fire happens, and we can come up with two of our state points at 3 and 6 by analyzing the regenerator because we know the of our regenerator. Then going from one to two is going to involve the pressure ratio and our isentropic ideal gas equations, going from four to five is going to use the pressure ratio and the isentropic ideal gas equations as well. So that should be enough to populate all the temperatures, which because I'm using the cold air standard is going to be enough to calculate the work in, the Q in, the workout, and the Q out, which will allow me to determine the network out and the thermal efficiency of the engine. So again, that's due to the fact that the problem told me to assume the cold air standard. If I wasn't using the cold air standard in this problem, I would have to look up PR1 and then use the fact that PR2 over PR1 is equal to P2 over P1, because that would be our isentropic ideal gas equation without the cold air standard. Then I would use PR2 in the tables to look up whatever I needed at state two, including but not limited to the enthalpy at state two. And I could calculate what my enthalpy would be at four by using the tables and the temperature. And then I would look up the PR4 as well, use PR4 over PR5 is equal to P4 over P5, again isentropic ideal gas equations and look up the enthalpy and probably the temperature corresponding to state five. And then I would set up my energy balance on the regenerator and use the effectiveness of the regenerator to determine H6 and H3, at which point I would have all of the enthalpies required to determine the work in the Q in the workout in the Q out, and therefore network out and thermal efficiency of the engine. Do you follow all that? Cool beans. So for my isentropic ideal gas equations from one to two, I'm using T2, that's a highlighter, it's unhelpful. I'm using T2 over T1 is equal to P2 over P1 raised to the K minus one over K. That again, came from our isentropic ideal gas equations when we've made the assumption of constant specific heats and other pressure ratios describing the pressure ratio across the compressor. Therefore P2 is 10 times higher than P1. So I'm plugging in 10 in place of P2 over P1. Therefore I can calculate T2 by taking T1 multiplied by 10 raised to the K minus one over K. We know T1, we know the pressure ratio, all we need is K and we can grab K from table A20 in our textbook. The cold air standard says to assume constant specific heats at 300 Kelvin. So we are using the properties at 300 Kelvin, which are CP of 1.005, a CV of 0.718, those are in kilojoules per kilogram Kelvin, and a K value of 1.4. So our temperature at state two, if the calculator would agree to help us out here, is going to be 300.15 times 10, 10 raised to the power of 1.4 minus 1 divided by 1.4. And we get a temperature at state two of 579.499. So I'm going to be jumping back to this page and writing down 579.499. And I could probably run that to 580 or 579.5, but we are going to be arbitrarily precise for the moment. So we got it from one to two and we can't get to three yet because we don't know five. So instead we have to jump to four and then analyze the process from four to five. Because I wasn't given an isentropic efficiency of our turbine, I'm assuming it's 100% efficient like the compressor. Therefore I'm using the isentropic adiogas equations again, which means we are going to be taking T5 over T4 is equal to P5 over P4 raised to the K minus one over K. We know P5 over P4 that's going to be one over 10 because we are going from the high pressure to the low pressure. So the after the turbine state point divided by the before the turbine state point is going to be a low number divided by high number, which is one over 10. So that would be 1400 Kelvin multiplied by one over 10 raised to the power of 1.4 minus one divided by 1.4.  And again, arbitrarily precise my decimal points there. So we have four and five, we have one and two. In order to come up with three and six, let's write out what we know about our regenerator. I'm going to be saying the amount of heat exiting the hot side, key region is going to be equal to the amount of heat entering the cold side, which is also key region. And remember, from our discussion of the regenerator by itself, I'm going to be characterizing key region as H5 minus H6 from an energy balance on the hot side. And the amount of heat entering the cold side is H3 minus H2 from an energy balance on the cold side. And then because of the cold air standard, meaning that I'm assuming constant specific heats, I can plug in Cp delta T and place a delta H. So I can write Cp times T5 minus T6, and that's equal to Cp times T3 minus T2. And then because the Cps are evaluated at the same temperature, I can say T5 minus T6 is equal to T3 minus T2. I have one equation and two unknowns, because I don't know three and I don't know six. The other equation is going to come from our definition of our effectiveness of the regenerator. We know that that's 0.8 from the problem statement. And remember that that describes key region over key region max, which I can write out as H3 minus H2, or I could have written out H5 minus H6. And then I'm substituting in T3 for T5 because under maximum heat transfer conditions, the temperature at 3 and 5 is going to be the same, and the temperature at 2 and 6 is going to be the same. So instead of plugging in H3, I'm plugging in H5. Because remember, for an ideal gas, like in the air standard, the enthalpy is only a function of temperature. So the temperatures are the same, the enthalpies must be the same as well. And then because of the cold air standard, the assumption of constant specific heats, plugging in CP, delta D in place of delta H, and then they are evaluated at the same temperature, which is 300 Kelvin, so I'm canceling the same CP. Therefore, 0.8 is equal to T3 minus T2 divided by T5 minus T2. That gives me my other equation from which to calculate my two unknowns. And in fact, by writing it out this way, that is 3 minus 2 over 5 minus 2, I have managed to write it in terms of only one unknown, and that unknown is state 3. So I can use this to calculate T3, which I will do by performing algebra, hopefully anyway. So I am going to say T3 is equal to 0.8 times the quantity T5 minus T2 plus T2. So calculator. Here we go. Let's do this. 0.8. I'm just remembering that it said 80%. I'm going to double check that it is indeed 80%. Multiply by T5 minus T2, which is that 725 number minus that 579 number. And then I'm adding to that T2. Yes, indeed. And I get 696.001. And that was T3. And then I can solve this equation for T6, which would be T5 minus T3 plus T2. And I now know 3. So that's going to be this 725 number minus this 696 number plus that 579 number. Let's just double check that. 725, 696, and 579. Yep, it seems right. Therefore, T6 is 608.624. Another way of double checking that I did that correctly here, which probably isn't as relevant to you guys since you're presumably looking at a single sheet of paper all at once as opposed to a really tiny zoomed-in section. Another way to double check this is to double check that your delta T from 2 to 3 is equal to your delta T from 5 to 6. So from 2 to 3, we are increasing the temperature by, what, about 120 degrees? That's 120 Kelvin as well. So from 5 to 6, the temperature should drop by about 120, and it looks like it is. So that's a good sign that we use the right state points. Now armed with our temperatures, we can calculate our work-in, Q in, workout, and Q out. And then from that, we can determine a network out and a net heat transfer in. We can look at them and go, ah, they're the same number. Hooray, we did it correctly, and then determine a thermal efficiency. So work-in is occurring between 1 and 2, so an energy balance on the compressor is going to simplify down to specific work-in is equal to H2 minus H1. Q in is occurring between 3 and 4. Remember that the heat exchange within the regenerator does not cross the cycle boundary. So an energy balance on the combustion chamber is going to simplify down to specific Q in is equal to H4 minus H3. Work-out occurs in the turbine, and energy balance on the turbine is going to be H4 minus H5 for workout. And energy balance on the Q out box yields H6 minus H1 for Q out. With each progressive example problem that we complete together, I'm going to be going into less and less detail about where these equations are coming from. If you've been following along, you probably have a good idea of how to come up with these. And in fact, you should be pretty good at it by now. It might be worth just practicing some drills, looking at a cycle, and trying to write out these equations before you even get to the state points themselves. It's a good skill set to have. Then because of the colder standard, I'm plugging in Cp delta T. So this would be Cp times T2 minus T1, Cp times T4 minus T3, Cp times T4 minus T5, and Cp times T6 minus T1. And I'm starting to run out of space here.  So Cp, again, from table A20 is 1.005. So we are using 1.005 multiplied by, calculator comes back, cswc. Thank you for your input calculator. We are using 1.005 times 579.499 minus 300.15 for work in. So work in is 280.746. And then 1.005 times T4, 1400 minus T3, which was 696.001. And that's 707.519. I'm going to write those down before they get lost. 280.746 and 707.519. And then I repeat the process for work out in Qout. It's 4 minus 5. So 1.005 times 1400 minus 725 ish. I'm going to work out of 678.248 and Qout is going to occur between 6 and 1. So 608 minus 1 yields 310. So my workout was 678.248 and Qout was 310.017. These are all kilojoules per kilogram. You know what's happening next. I want network out, which is going to be work out minus work in and net heat transfer in, which is Q in minus Q out. So I'm going to take 678.248 minus 280. And I get 397.5. And I'm going to take 707.519 minus 310. And I get 397.502. Parade and the same. That's a good indication we built those equations correctly. And we didn't plug in any incorrect numbers. At least if we did, we did it very consistently. Then I can determine my thermal efficiency by taking the network out over Q in, which is going to be 397.502, this number, divided by Q in, which was 707.519. And I get 0.5618 or 56.18%. And I was looking for this and this as my answers. Then I asked us to compare that to the first two brain cycle examples. So I will copy over those results. And then we can look at them at the same time and go, ah, look, two was lower, three is higher. Interesante. So we got 397.5. We got 56.18%. So in comparing these, I see that the presence of isentropic efficiency has decreased my thermal efficiency and my network out. And remember that the presence of the regenerator will decrease how much Q in is required for the same amount of network out. So it doesn't surprise us that we have the same network out because we have the same maximum temperature. So we're decreasing how much Q in was required. Therefore, we are increasing the thermal efficiency. So the presence of the regenerator is going to increase the thermal efficiency. And remember, even a terrible, terrible regenerator is going to be improving our thermal efficiency. So hopefully we can find a good sweet spot between spending more money on a heat exchanger that works pretty well, because theoretically to achieve 100% effectiveness, we would need an infinite amount of surface area, which is hard to do with just fins and more surface area. And that investment of money is going to mean that we're paying less money in fuel, and we could probably spend a little bit more money on our turbine and compressor to try to get a slightly higher isentropic efficiency. And it would be your job in this situation to calculate what the breakeven point is. Where is the threshold of spending more money on a regenerator or a compressor or a turbine isn't actually paying off in the long run using projections of fuel prices?