Good evening to this evening's worship. We are welcoming you to the evening worship. We are grateful that you have joined us today. Before we begin the survey, I would like to ask us to bow our heads free. Dear God, we thank you so much for the gift of life that you have given us. We thank you that you have been with us through the day. We have come to the evening part of the day where we want to take a few moments to feed, to worship with you and to pray with fellow believers. We want to ask you to bless our hearts and open our minds that you will be able to listen to your word, receive it, keep it in our hearts and also be able to share it with you. I pray this is a slim. Again, welcome to the evening worship today. And I think I'll just introduce myself. My name is Ebi Okoku. I will be the chorister for the evening. And the pianist is Mrs. Opio, Lawrence Opio. I don't know how you'll say hi. Amazing. So we'll begin with a few items this evening. So let's just turn our song, our hymn, to our hymnals. Song number 545. Saviour like a shepherd. For those of us who are in church, I'd like to ask you to stand so that can let us pray. Our kind and heavily father, we want to thank you for the gift of life and heavily father for this wonderful week where you have brought us to learn about your secrets of prophecy, heavily father. Three-and-seven, as Thar'jau whom you have used most heavily father to present this wonderful message to your children. Continuously use him heavily father and may this message impact our life and everyday. Bless your listeners who are listening to you throughout all corners of the world that heavily father. This message may be able to provoke them heavily father, may may carry out your message so that you soon return to be able to come. this day till the end. All this we ask the name of Jesus Christ for Lord and Savior. Good evening brothers and sisters.  We thank God for this opportunity once again that we have come to worship him. We in a special way would like to welcome each one of you who is joining us here in the sanctuary. Do you feel blessed? Do you feel a bit of Jesus? And also in a special way we would like to welcome our congregation following us online. Feel much welcome in this house even as we continue to listen from the servant of God on this day six of this secret presentation. On behalf of the pastorate leadership, whole church leadership, may you all feel blessed and may we all find a place a Jesus fit that you may learn and may we grow closer to him. May God bless you all. Yes, good evening everyone. We welcome you to the Divine Hour and our scripture reading for today is coming from the book of Isaiah chapter 54. A read from verse one to two. It says, Amen. Stand to the opening song. Team number 439. I request the congregation present to rise up as you sing the song. And place you for your brother. Thank you for giving us an opportunity to sit at your feet. And here the wonderful plan you have had for humanity tried all ways to reach East China. Thank you for the way you use the prophets of all so that this message can reach us. We commit ourselves to you as we continue with our series this evening. We pray in Jesus name. Welcome once again to our prophetic series, Secrets of God. And when I read the book of Isaiah, I marvel at the masses and love of God and how he informs his people in advance what he intends to do and why he's planning to do it. We have seen that this wonderful book of Isaiah has two streams of prophecies. The cardinal stream, the main stream, the stream of salvation, the prophecies of salvation. All the way from 740 BC to the time God created the heavens and the earth. I just want you to know that Isaiah goes past the head of the world. When you come to Daniel, we shall see that Daniel reaches the coming of Jesus. He doesn't go further. But Isaiah is the one who goes farthest into the future, past the millennium which he does not mention until God will create the new heavens and the new earth. But there is also the second shortest stream of prophecies which he pick up at yesterday, the deportation of his people because of rebellion. Let me just tell you in my studies of the Word of God and it's an experience. I'm of the opinion that God has suffered more from his people than from those without. Because Gentiles are there, they don't know God, they don't know about him and therefore they go their way. But here is a people who have identified themselves with God. They are called by his name but they are his greatest troublemakers. It depends on God. The goodness of God is that he does not leave them. He struggles with them and that's why God had to take his people whom he had chosen from among all the nations. He took them to captivity and it depended on God to lose ten tribes. It depended on God and in fact we don't know how much God has suffered in the hands of his people. But Judah was no better. The loss or the destruction of the northern kingdom was not that deterrent to Judah.  That's why there is imprisonment and other kinds of punishments. But here Judah sees the northern kingdom lost but they don't learn. Until now they also go into captivity but by the by the masses of God he redeems alumni and brings them back. And I told you that when God, when his people moved to Babylon, he also moved with them. He relocated to Babylon for 70 years. I'm just going to read you one wonderful text by Isaiah on this fact of God relocating to Babylon. As Isaiah was talking about the coming of the Messiah, the repatriation back to their Lord, this is what he says in Isaiah 52 verse 8. Listen you are watching men lift up their voices together they shout for joy. When the Lord returns to Zion they will see it with their own eyes. It's clear that there will come a time when the people in Jerusalem, as they are there in Jerusalem, they will see the Lord returning back to Zion. I told you now because these people have gone back to Jerusalem, he also comes back to continue with the preparation for the first advent of the Messiah. Having done that yesterday, now we go back to the old story, to God's international agenda, the Salvation of Manikai, so that we continue with the story from where we left. You remember we left the story where the Messiah came in Isaiah 53 and through his death and the Salvation, through his death and the Salvation, he saved the human race. Now it is very sad for that Isaiah 53 is not read by other dogs, Jews. In fact by Jews in common. But for the other dogs, Jews, they have even put a curse on any Israelite who would read Isaiah 53 and therefore Jews don't read this chapter. They don't understand who is this person. Let me take you back and tell you that Isaiah gave two prophecies concerning the coming of the Messiah. The first one said, Isaiah 714, a child who would be born, whose name will be Manuel. The second Isaiah 9 verse 6 and 7, the mighty king, the everlasting father upon whom the government will be. Now you can see one is talking about God with us and the other one is talking about the Messiah coming as a king as a ruler. Now the Jews took Isaiah 9 at the expense of Isaiah 7 and therefore they have been waiting for a kingly Messiah. The one who is going to sit on the throne of David, the one who is going to rule the nations, the ones who is going to make Jews rulers because according the Jews what they have, their concept about God's kingdom is a political kingdom. But God was preparing a spiritual kingdom and therefore they did not understand Isaiah 53 and Isaiah 53 has benefited the Gentiles more than the Jews is very unfortunate and that's why Paul is crying in Romans 9, Romans 9, 10 and 11 trying to answer the question why didn't my people reject the gospel? Why didn't my people reject this wonderful, it was for them but that's how man is. And therefore the Messiah died and saved the human race. Now today I won't answer the question what was the outcome of the Messiah's death and resurrection. What was the outcome and to that we now go in beginning with Isaiah 54. We heard of Isaiah 53, the prophecy of the coming of the Messiah and his work of redeeming mankind. And let me tell you who is listening to me, you have been redeemed in Isaiah 53, God has removed the barrier that was there between you and him. And that's why Paul is saying in 2 Corinthians chapter 5 verse 18 to 20 that God was in Christ reconciling the word unto himself, not imputing their sins to them. And he has given us this ministry of reconciliation. Therefore God has reconciled the human race to him. He has removed the barrier that was there. This was God's work, God and his son. Those who are the beneficiaries don't know it, they do not know it. How are they going to know it? Is God going to implement this wonderful plan that has been executed by his son? That's why all the three members of the Godhead have been involved in your salvation and my salvation.  And now the third member of the Godhead had now to implement package those who are the beneficiaries. And that's why Jesus said in John chapter 16 verse 7 down once he said John 16 verse 7, but I tell you the truth, it is for you are good that I am going away. And as I go away the counselor, the comforter will not come to you. But if I go, I will send him to you. Verse 8, when he comes, he will convict the word of guilt in regard to sin and righteousness and judgment. In regard to sin, because men do not believe in me. In regard to righteousness, because I am going to the Father, where you can see me no longer. And in regard to judgment, because the Prince of this world now stands condemned. That was not the mainst of the Holy Spirit.  Your destiny is eternal death.  All you need now is to accept God's gift of salvation. That was the work of the Holy Spirit and he was going to use now the new instrument that now God has commissioned after the failure of the Jews. The new instrument that now God has commissioned is the church. And that's why Christ before he died, he said in Matthew 16, verse 18, that I'm going on this rock, I'm going to build my church and the gates of hell shall never prevail against this. Jesus is saying, Israel are no longer God's instrument to bring about salvation to the human race. They failed for 1,500 years. God has struggled with them and he failed. They failed him. Therefore, because I have saved the human race, I'm instituting a new instrument, the church. And I know the church will be battered by the world, certainly will fight the church, but the church will triumph for gates of hell cannot prevail against the church. Therefore, two entities to implement God's plan, the Holy Spirit using the church, that was had been prophesied by Isaiah because this is the outcome of the death of Christ for our sinners. And therefore, when Christ died, he produced what we call the gospel, the good news. And now according to God's plan, although he knew that the Jews were to reject his son as the Messiah, they were going to be the first beneficiaries of this plan. Do you know there is a statement by Paul very significant in Romans when Paul is introducing his book of Romans, he gives it keen notes in verse 16. And he says, I want you to listen. I am not ashamed of the gospel because it is the power of God for the salvation of everyone who believes. Note that the gospel is the salvation is the power of God for the salvation of everyone who believes none is exempted. Salvation is an inclusive. But now note the next words. First, the Jews, then the Gentiles. Why did Paul say that? Paul is saying although the plan of salvation is to save the whole human race, it must first of all go to the people whom God had chosen, the Jews. And this is how the message went to the Jews. Isaiah 54. Isaiah 54. The message of salvation to the Jews. Now that there is a gospel to preach, now that there is salvation to give to sinners, the Jews who rejected the Messiah are going to be the past beneficiaries. Now I told you that Isaiah is written in a poetic way, in a poetic form with most of the message in a metaphoric form, in a metaphor. Listen to the message to the Jews in Isaiah 54. Sing, O barren woman, you who never bore a child, burst into song, shout for joy, you who were never in labor because more are the children of the desolate woman than of her who has a husband. Isaiah 53 has saved the nation of Israel without them knowing, they don't know it. And now God tells Isaiah, tell the world that the first message will go to them. They are going to be the first beneficiaries of the plan of salvation. The gospel saves everyone, but first the Jews and then the Gentiles. And now Isaiah 54, the message goes to the Jews. And the message is, you, bed, and woman. What does that mean? What's God saying here?  You never gave any. You are barren. As far as giving me children is concerned, you never give me any children.  And therefore you are burdened as far as I'm concerned because you never bore me a child. God is telling the Israelites, I wanted you to proclaim to the Gentiles that our God is our Savior. He can also save you. But instead of you, Jews, using my temple to bring in the Gentiles, you made it exclusive for you people. And you made the rule, no Gentiles should come near my house of worship. That's why the Gentiles don't know His. They don't know what I've done to them. You never bore me a children. But now, burst in song, shout for joy. You who never went into labor to bring a child, to bring a baby. Why? Because what you've failed to do, I have done it. Hallelujah. What you failed to do, I've done it. And you are the first beneficiaries. What kind of a God do we worship? He knows all the mess that I've done, but He makes them the first beneficiaries. Shout for joy. Enjoy the salvation that my son brought to the world. Although you killed him, you didn't know that by giving him to the Gentiles, to the Romans, you are facilitating your own salvation. Now, enjoy it. Shout for joy. The message of salvation should not be to make you gloomy. It's a message of joy. And that's why Paul is saying, I'm not ashamed of the gospel. And he goes on going to be happy. Why are you going to shout for joy? Because your children are going to be more than a woman who just had two, three children for the husband. Your children are going to be in millions. They are going to be so many. Therefore, although you never gave me children, I got the children. I gave you the credit. Shout for joy. Jesus. And we are going to see tomorrow evening how this happened in actuality, because this is prophecy. How it was fulfilled. Now, they were to shout for joy, because as God was looking at the world, the millions who were going to accept the gospel, me and you included. Hallelujah. God could tell. Can you just see the joys and sorrows? My program has not failed, but you have a work to do. Is there? Yes. Enjoy the benefits. Shout for joy. But there is a task for you to do. Verse two. Enlarge the place of your tent. Stretch your tent curtains wide. Do not hold back. Lengthen your cords. Strengthen your sticks. For you will spread out to the right and to the left. Your dissidents will dispossess nations and settle in their desolate settings. Yes. Is there a joy for what God has done? But there's a work for you. Extend any large your tents. Expand your tents. Why? Because of the influx of the people who are going to come. Now, this metaphor is taken from the Jewish culture where whenever our family had a big occasion where visitors are going to come, you see every family had a tent for the family. Just attend that he feels the family. What about when the visitors come? When the visitors come, it was not customary to bring to go and beg our other tents so that you bring for the visit. All what the woman did is to remove the pegs from where they were for the family and extend it further. Are you getting it? Extend it further so that that can be a bedroom for those who have come. And therefore, every Jewish family had a tent that couldn't expand and God gets that metaphor and tells Israel because I am bringing many enlarge your tent. Hallelujah. Expand. What is going to tell Israel is telling them. My people, you have been too exclusive. You have contained me as your God. I'm not a tribal God. I'm the universal God of all the people. Don't contain me in your theology of exclusiveness. Enlarge the tent. Expand the tent. You do that work. I'll do the work of bringing them. Hallelujah. Your work is to extend the term. What is telling them? You Jews refrain from your exclusivity because you have been, you have been against my plan without knowing by making your religion too exclusive. Now, the Jewish religion, although it was meant for the world, was turned by the Jews to be theirs. And do you know what the Jews believed? There's nothing good you should do for the Gentiles, for they were created as fuel for hellfire.  Terrible kind of a prayer. In fact, after now I've not discovered what was the prayer of the women because this was the prayer of men. I praise you God. You never made me a Gentile. You never created me a Gentile. They had nothing to the Gentiles. But God has saved Gentiles because the first prophecy to Abraham was that through you, Abraham, all the nations of the earth will be blessed. They heard that scripture. It is in their Torah, but they did not understand it because of what we call their kingdom theology, a theology of power we want to rule the Gentiles because they were made created for hellfire. They are for the message, for the Jews was to leave Judaism to understand that Judaism has been a barrier to them. Abandon it and join inclusivity where all the people will be included. That was the message of Isaiah chapter 54, the outcome of the good work of the Messiah, Israel. You whom I commission to work for me as you deserted the work. I have done it. Enjoy the work. But leave exclusivity because I'm bringing an inclusive religion. I'm bringing a new chapter. I have to be worshiped. Remember Isaiah 2? I have to be worshiped by the whole world. That was the message to the Jews. God was telling the Jews, remove the war of partition. You see, by the time Jesus came, it was very hard. Nor were Gentiles to worship in the temple. In fact, they had put different protocols for the worshipers. That is for women. That is for men. That was for the priests. In fact, here was the sanctuary. Next to it was the one for the priests who were nearest to the temple. Then the next were men, were women. Then men. And after the men, after the faradah for men, they put a wall. And it was written at the book that any Gentile who will go past this wall would be responsible for his death, which will ensue immediately. That is the kind of religion that Jesus found here. That Gentiles could only see the temple from far. Because there was a wall. And that writing was in three different languages. It was in Greek, in Roman, and in Aramaic, the Jewish language. And therefore God was telling them, I'm going to bring my people. And you should now open the gate. You should open the temple. You should open the spiritual temple. And let me say, sometimes even we Christians, we have a sort of exclusivity. Although we go missionary, we still believe that this is ours, the others have theirs. Now God is for everyone. Now having brought, having given the message to the Jews, now the Jews were to rejoice in the message of the gospel. And they were to enlarge the tent for the infrax. Now God left the Jews. He has preached to them. He has invited them to the plan for which they never walked. God is telling them, enjoy it. Enjoy its fruits. And its fruits, one word, justification. Justification by faith. In credited righteousness that was credited to Abraham in Genesis 15 verse 6 when he believed. Jews, all I require from you is to believe in the Messiah you rejected and get the benefits of justification by faith. Having done that in Isaiah chapter 3, now God returns with the Gentiles. You remember Paul saying, first to the Jews, then to the Greeks. The Greeks represent the Gentiles. That's where now we go to Isaiah 55, Isaiah 55. The message to you and me. You see, Isaiah had prophesied that this message will come to you. They want to know, to know there's a difference. This is an invitation to the Gentile world. Isaiah 54 is an invitation to the Jewish world. Come, all you who are thirsty. That's how God opens the message to the Gentiles. Come, all you who are thirsty. Come to the waters.  Come, buy and eat wine and milk without money and without cost. Why is paid money on what is not bread and your labor on what does not satisfy? Listen, listen to me and eat what is good. And your soul will delight in the richest of fear. The message to the Gentiles. The message to me and you. Come, all you who are thirsty. This is an invitation to all. Because Isaiah chapter 3, in verse 6, had included all. All of us have gone astray. Each one has gone to his own way. But the Lord has laid our iniquities on him. All. And because all the iniquities of all have been laid on the sun and he has paid the penalty for them by his death. Now the invitation goes to all come because I know you are thirsty.  He is dust of righteousness. He may be okay in all other areas. But when it comes to righteousness, he is a dusty person and every sinner is a dusty person. He may know it or he may not know it. And God knows the extent and the depth of that dust. And you know, God is not saying hunger. Because you can continue with the hunger for several days. I remember when someone was dark, was rescued in that pit where they were digging for God, I think. I think after six days, after six days, you can see the word of God. Now, dust does not go for long because the body will be dehydrated and you will just die.  Therefore, God is opening to sinners. Come, you drink. You remember Jesus, you are the same during the come meeting in John chapter seven when he called the people, come to me all you are thirsty and drink. The same metaphor. Now, then he goes and says, come, buy wine and the milk without money and without cost. Another metaphor, another symbolism. He is telling people, come and you buy. But I don't want your money. Now, we know that in our modern world, all transactions are done through money. Money is very crucial. In fact, when this pandemic came, the first thing the government had to handle was how are people going to handle the transactions because life has to go on. And you remember, they said that you use the emplacement. And very interestingly, for two years, I have not touched the money and I will not have any transaction with you through hard cash. I use emplacement. Muah told that money has gone through so many hands, but God is inviting Gentiles to come and buy, but without any money. What does that mean? You know, before our modern business transactions, people were still trading. Even before we were colonized, even when we were only Africans, we were trading between ourselves. And what people did in their trade was called butter market. Now, in the butter market, you don't have money. What you do is that you exchange what you have with what you don't have. And mostly in the reserve, when there was hunger, what the people did is that they took what they had or God had blurred them with. If it is millet, they go to the other tribe who did not have millet, but they had emice. And therefore, they agreed to the transaction. So many did not have the weighing machines, and therefore they used containers. So many karabaches of millet will be equal to so many of maize. And you give the millet and you get the maize. This is what you did not have. This is what you wanted. You give what you had, you get what you did not have. That was the butter market, the butter trade. This is the one God is using here.   I have eternal life. I have forgiveness. I have justification. Give me what you have, but I'll give you what I have. We call it the great exchange. Hallelujah, saints. The great exchange. Do you know that the salvation of a sinner is an exchange between God and the sinner? And God is telling the sinner, give me what you have. And what does the sinner have? All he has are sins. Nothing else. A sinner has nothing else to give to God. All he has are sins, dreadful ones that are taking him to hell, to eternal oblivion.  I'll have them and I'll give you my righteousness. That is the exchange being stabilized here by having, by buying something without money. And this transaction is well articulated by Paul in second Corinthians, chapter five. Second Corinthians, chapter five, listen to the transaction. Verse 18, all this is from God who reconciled us to himself through Christ and gave us the ministry of reconciliation that God was reconciling the word to himself in Christ, not counting man's sins against them. Why? Why is God not counting their sins? Because he has counted them on Jesus Christ. And he has given us the message of reconciliation. Verse 20, we are therefore Christ about us as though God were making appeal through us. We implore you on Christ's behalf be reconciled to God. Why?  God made him who had no sin to be seen for us so that in him we might become the righteousness of God, the better transaction. Jesus took our sins and gave us his righteousness. Now as we close our message of Isaiah, 55, he goes on and says, University, seek the Lord for he may be found, call on him for he is in there. Let the wicked forsake his way and the evil man, his thoughts, let him turn to the Lord and he will have mercy on him and to our God for he will fully pardon. Wonderful message to the Gentiles is seek the Lord for he may be near. Why he's saying that? He's telling the Gentiles, there was a time God was very far from you. And Paul is telling the Gentiles that there was a time that you are not a people. God was too far away from him. In second, when he was talking about innovations, chapter two verse 13 he says that listen to this, remember that from verse 12, remember that at that time you were separated from Christ, excluded from citizenship in Israel and foreign us to the covenants of the promise without hope and without God in the world. But now in Christ Jesus, you who once were far away have been brought near through the blood of Christ. That's the message.  Now seek the Lord when he's available. Since the death of Christ, God is near every sinner and all what the sinner is being told to do seek the Lord when he may be found. This is the time to seek the Lord for his near. Since the death of the Son and the cross, God has been near every sinner without the sinner knowing and he's just telling the sinner, please seek me, accept me as salvation is you. That is the message of Isaiah to the world and therefore this wonderful messenger of God proclaimed everything that God was to do and therefore the New Testament is just a fulfillment of the promises of Isaiah, of salvation. The Messiah died and I serve everything. The gospel went to the Jews, I serve 54 and then it has now been going to the Gentiles and even during this Quinequenium the message is that I will go to do what, to tell the Gentiles come and have a better trade with God. Give him your sins and you get his righteousness. I will add by a statement by one of wonderful scholars on the book of Isaiah. Listen to what he says. These three chapters, I serve after three, 54 and 55 form in themselves one of them of those significant riots that characterize our book of Isaiah. The first chapter 53 gives the groundwork in the sufferings of the cross. Then in chapter 54 the efficacy of those sufferings announced to Israel and now in chapter 55 we come to their worldwide proclamation. Amen. He has come to you just as Isaiah has prophesied for you to get it and to give it to others. It made me so. It's our prayer in Jesus name. That's our prayer. That's our prayer. Thank you for the gospel to the Jews to rejoice in what they never worked for and to launch their tent for the infrax of the Gentiles. Thank you for the message to the Gentiles to come without money. Poverty is not an issue. You just come and do a transaction with the God for you have sins and he has righteousness. Have an exchange program with him and heaven is yours. May it be so forevermore. It's our prayer in Jesus name. Amen. Let us sing the closing song in number 439. How far from home? Can we rise? Present your sister for that. I want to thank you all for tuning in to our program this evening. The Lord has been with us. We have received blessings from the man of God Pastor Francis Jiao on this series of prophecy. I want to thank you all those who are watching live from your homes and some of you in your offices, whatever you're watching us from Facebook or YouTube. We thank you so much for choosing Narobo Central Church via source of your blessings even as we journeyed through this week. Just to inform us that this series is also in one of Pastor's books and you can access those books from our ABC bookshop right here in Narobi City if you are from within this city. So you can walk to our ABC and you can have a copy of the series of these prophecies of presentation that is captured in one of his books. This coming supper on 18th we will be having baptism and we are appealing to anyone out there who has been waiting to be baptized. This is your time. Please drop your name to any of the baptism classes, leaders or to our pastors or any of the elders that you may be prepared for the same. Our programs continues tomorrow at the same time from 5.30. We invite you to be part of the program. Tell our friend to come and fellowship together with us. May the Lord bless us. Let's pray as we close the fellowship this evening. Gracious Father in heaven we thank you for being so kind and gracious to us even for giving us this privilege of this fellowship. We thank you for the messages that have been coming to us every evening Lord. We thank you for speaking to us through your man servant. I pray that we all may see you and find you and we may have salvation. We pray Lord as we come to the end of this week that we will be revived, we will be equipped, our eyes will be opened. We'll be able to see the world and salvation in the light of heaven that we may be fully prepared for the soon coming. Those of us here in church Lord I pray even as we journey back home you can give us a safe journey and those at home Lord continue blessing them and that all together we will be blessed of you and prepared by the Holy Spirit for the soon coming. This is our prayer this moment we pray in Jesus name amen. God bless you all. We meet tomorrow at 5.30


I was going to do a visual gag about Graham's number, but I've only eaten a Googleplex of these things and I don't know if I can go through with it. There's a branch of mathematics known as Ramsey theory, which is concerned with finding the limits on how many things you absolutely need to satisfy some criteria. For example, if you arrange a number of dots into a circle and then connect them all with either red or blue lines, how many dots do you need minimum so that no matter how you end up choosing line colors, you can't avoid drawing either a red or a blue triangle? Questions like these piqued the interest of mathematician Ron Graham, who tried his hand at a slightly different problem. Rather than circles of dots and red or blue triangles, he was thinking about dimensions of cubes and red or blue this thing. How many dimensions does a cube need to be before no matter what, you can't avoid drawing this figure in all the connections between its corners, either in all red or all blue. In a two-dimensional cube, which is just a square, it's pretty easy to avoid. Just use a different color for one of the edges. Duh. In a three-dimensional cube, which is just a cube, it's still pretty easy. You've got double the number of corners and a bunch more edges to worry about, but with a little bit of thought, you're still probably fine. Now, if you're anything like me, you're probably not used to thinking in four dimensions. But moving from 3D to 4D is very much like moving from the square to the cube. You double the number of corners again, so 16 now, and then you connect everything to everything else. We're now looking at 120 edges, any one of which could be colored red or blue, which means that we're looking at two to the power of 120 different colorings of this figure. That's a number that's 36 digits long. But computers and mathematicians are pretty good at this sort of thing. And it turns out that in four dimensions, we can still avoid drawing this figure. We can avoid it in five and six dimensions all the way up to twelve. So can we just keep on going forever? Well, no. Graham didn't find an answer for how many dimensions we absolutely need before we have to draw this figure somewhere inside the cube. But he did find an upper limit, a number that's definitely bigger than the one that we're looking for. In fact, it's definitely bigger than the vast majority of numbers that we use. It actually held a record for the largest finite number used in a productive mathematical proof, and it's not hard to understand why. Mathematicians have invented some notational tools to represent stupidly large numbers in a way that won't take all day to write out. This up arrow is used in Graham's proof, and it denotes a sort of recursive power of exponents. x up arrow up arrow y means raise x to the x-th power y times. So two up arrow up arrow three means that you get a stack of three twos, two squared squared, which is just two to the fourth power, or 16. Now, if you're crazy enough to add a third arrow, you're saying raise x to the x-th power x double arrow y times. So two triple arrow three means make a stack of sixteen twos. If you've got a second, pause the video, find a calculator, and try squaring two fifteen times. That's a pretty big number. Let's say it's just much larger than the number of particles in the observable universe. Graham defined a number, G1, as three quadruple arrow three. That's a three with three triple arrow three exponents on it. If you were able to divide the entire observable universe into Planck lengths, the smallest meaningful distance in quantum mechanics, and then somehow fill each of those locations with a single binary digit, essentially making a universe-sized hard drive, you'd have around 10 to the 180th power digits to work with, and you still wouldn't even come close to being able to write down G1. But that's not Graham's number. That's just where we're starting from. G2 is three G1 arrows three. G3 is three G2 arrows three. So on and so forth up to G64 Graham's number, which is the upper limit to this problem that he discovered. So the answer to the original question is a finite number somewhere between 13 and Graham's number. It used to be between six and Graham's number, but mathematicians have managed to close the gap a little bit. As I've mentioned in previous videos, there are all sorts of clever ways that we can dissect large numbers to try and interpret them meaningfully. Like an aircraft carrier is about three football fields long. It has a crew about the size of a small town, that sort of stuff. But G64 is too freaking big for us to pare down so that we can handle it in any way. I mean, there is that limitation of the size of the universe, but perhaps more appropriately, there are limitations of psychology too. In that episode, I cited some evidence that we used different neurological systems to crunch numbers and to estimate answers. The system that we use for doing math, for pushing symbols around counting and following mathematical rules, is very closely associated with the parts of the brain that are normally used for language, while rough estimation of answers and figuring out what those symbols might practically mean, is more closely associated with the parts of the brain that are used for visual memory. It turns out that isn't the only split that happens in our heads when it comes to numbers. Our estimation apparatus has a couple of different gears that it uses for cognitive representation of quantity, and you don't need to get to Graham's number for them to start doing something weird.  A lot of research has gone into trying to understand the details of how number sense works. Most researchers agree that it's composed of two distinct systems, one that's good for small exact numbers and the other that's good for moderately large, inexact numbers. They both have some very interesting and unique properties unto themselves. Fascinatingly, this isn't just a human thing. There are all sorts of diverse species, which exhibit some sort of number sense. Guppies, monkeys, pigeons, rats, ravens, dogs, and cuttlefish all exhibit a similar pair of psychological mechanisms at work with similar strengths and weaknesses. First, the parallel individuation system is concerned with quantities of one to four things, which it can track with ruthless efficiency. Whether you catch a brief flash of some assortment of stuff, or have someone poke a few fingertips, so long as we're talking about fewer than four whatever's, parallel individuation is on the job. It's called parallel because it allows for discrete tracking of each one of those objects at the same time. You don't need to count to know exactly how many cookies are on that plate. You can just look and know that there are three. 2. Koko? Of course, Graham's number is larger than four, so no help there. How about this other system? Well, the approximate number system works by grouping a quantity of stuff into a single mental symbol, like about 300. While the parallel individuation system can tell if one thing goes missing every single time, the approximate system has a bit of a buffer zone where it doesn't really register that anything's different. 260, 340, meh, who cares? It's about 300. And weirdly, the amount of change necessary for the system to register that something's different is logarithmic. It tends to exaggerate the differences between smaller numbers and kind of fall asleep for larger ones. Like going from $10 to $15 for a movie ticket, that feels huge, but hearing that someone's going from a $40,000 to a $41,000 a year salary, unless I sit down and really think about how much money that is, how many movie tickets, I barely register a blip on my mental radar. That logarithmic error scale does improve some with training. Children are more susceptible to this phenomenon at lower numbers than adults are, possibly because adults have some experience working with larger ones. But it's still present even in highly trained adults. The larger the numbers, the less we care about the same amount of change. Now, Graham's number is just so far outside anything that we might realistically encounter in the world that NumberSense just doesn't work on it. The best we can do is a numb feeling of incomprehensible bigness, which, although accurate, isn't really informative. But there are large numbers that we do encounter that NumberSense is nonetheless ill-equipped to handle. A 1 in 14 million chance of winning the lottery? A microchip that can perform 20 trillion operations a second? Even a $200,000 mortgage is just so big that we can't bring the same intuitive faculties to bear that we would use to decide between an $8 medium or a $12 large popcorn. The only recourse that we have to interpret these numbers in any meaningful way is to dissect them with mathematics, to break them down into quantities that the approximate number system can work with reasonably well without reporting back that the difference of $1,000 isn't really a big deal. And speaking of math, in 2014, a team of mathematicians came up with a smaller upper bound on the problem that Graham was working on. Two triple arrow six. That's just two to the power of four trillion or so. Chump change, right? Best I can do is two double arrow six. How's your NumberSense of Graham's number? Please leave a comment below and let me know what you think. Thank you very much for watching. Don't forget to blah, blah, subscribe, blah, share, and don't stop thunking.


Good morning everyone. My name is Tim Dryansdale and I am a senior lecturer at the Open University. And today I want to talk to you about something I'm calling LabRTC. What is it all about? The problem I want to talk to you about is if you've got equipment all the way over here and people all the way over here, how do you get them together? How do you connect people to your hardware? This is an open hardware event. It probably means, for the most part, you've got some super cool projects but what you want to do is get people connected to your stuff. And if we roll back to the 1950s, the way you do that is you say, gather round. Come stand at my bench and I will show you my, in this case, Trollus experiment. However, now we have super cool hardware, things like a Miarm, I've got one in my house. And we're kind of still doing the same thing with hardware. We're still kind of saying, come round and see my thing. But the problem is we've got the internet, we've got an audience that is far bigger than the number of people that can drive to your house to see your latest toy. So you can do stuff like web sockets, which actually we use, which is fantastic, but look at this, there's a big green expanse here where there should be a picture of a robot or something. There's just an on-off button for an LED that I took this example from. And the lack of video is kind of a pain because everything we do now is about having video. Did it happen? Well, I can't see the video, therefore it did not happen. So if you want to get publicity for what you're doing, if you want that publicity to get you money, then you're going to need to have a compelling experience. And with all of the recent burns that have happened to people who bought Kickstarter projects and prototypes haven't worked, you need to do everything you can to have a compelling interaction with your hardware so that people will trust and believe what you're doing. And video is part of that. I've got a personal reason for being interested in having connection to hardware from a distance, and that is that my parents live on the other side of the world. My dad's quite techy and sometimes I like to show him what I'm doing. So we've got two good reasons already. One is getting publicity for your project. One is me helping my dad see what I'm up to. And the other one is my day job, which is spending quite a lot of Hefke's money on making a room full of equipment with a locked door that I can't let any students into actually do something useful for us. So all of that gets gone in a room, no students are going in it, and I've got to get them connected up to it and having a realistic experience. And so we share the same problem when we think about getting open hardware connected to people and my students getting connected to people. Why did we go remote? The Open University is a distance learning university. We do degrees, but the students don't come on campus. We used to use the television to get lectures out there, but obviously things have moved on a little bit since then. But something that's only very recently started to change is the use of home experiment kits. We, just skipping over some of those details, have a new electronics course. And electronics you can't really do unless you're doing something with hardware. So for us to offer it, we have to move on a little bit from some of the approaches that have been taken with home experiment kits being sent out to students because we want students to interact with stuff that, quite frankly, is far too expensive to stick in the post and send one out to every student. So we have a new strategic approach overall at the university anyway, which is not to send equipment out to students. And instead that forces us to look at how students can access them remotely. And it's actually a really good thing because it forces a great deal of creativity and encourages us to think really hard about what should happen in a lab environment and what the learning experiences are. And the people at the Open University that have gone before me in doing this, Nick Braithwaite in particular, one of his team is here, Edward Hans, who is waving in the front. I'm disappearing shortly after this, so if you have any questions about what we're doing, fall upon Edward. So I promised I would publicly embarrass him by pointing him out at this stage. So before I arrived at the Open University, I already won an award for the way in which it engages with students over the internet using hardware. And we were recently funded again, a different fund of this time and as part of that I'm looking after doing massively parallel electronics experiments at a distance for students. We've got some robots in one room. We have some national instrument equipment in another. Obviously none of this is open hardware per se. So for practicality reasons it's what we use. But the things that I'm doing to connect to this equipment relate directly to what I think you should be doing to connect people to your open hardware. We have a giant switch with lots of purple wires coming into it that's directly connected into the Janet Fiber optic backbone and that is solely for experiments. So just to give you some idea of the scale of what we're proposing to do, it's not just one experiment and a big queue of students. It's lots of experiments and an even bigger queue of students but they're more using it in parallel. So how do we actually go about showing something off? Wow, you should stick a webcam on it. So that's fine. You can stick a webcam on something but that's when your problems start. The speed of light is actually not the problem here. Once you've got your video onto the network it can go all the way around the world in less than a reaction time, more or less. So the problem you get with sharing video is actually latency. And you can try and get ahead of the game by starting slightly before the gun by predicting in certain things, not in this particular case. But if you're doing video, he says, clicking, then there are a number of things that you can try. And one that we've used successfully at the Open University is JPEG Refresh which works if you've got your equipment behind lots of security but you still have a latency issue there. You can stream stuff. We were doing some experiments with some security cameras and because this is being videoed I must say that I've just pulled these images off the web. I don't actually know which ones my team were testing so I don't know if these are the ones that are included or not. But basically we found that if you wanted a streaming camera, you had to buy a security camera and then the cost for the quality of video you got wasn't really particularly a compelling, how should we say, cost-benefit compared to what you can get out of those Logitech C920s. So we were looking at spending hundreds of pounds of camera to get a decent video quality with this built-in streaming so it didn't seem economic to us. And then you might start asking, well, if you're looking at all of these different latency issues and you've got a few seconds over here or you've got three minutes over there or you're starting to get a sub a second, what really matters? How much latency do you tolerate before somebody says this is no longer a compelling experience? And in computer games the answer is whatever latency means that you don't end up with people thinking they've shot people when they haven't been. That's the biggest previous kind of look into network latency.  But that doesn't work for us because real hardware is only going to make it into our lab if we're doing an experiment that is too chaotic or random or difficult to simulate. So if we could simulate it, we'd simulate it anyway. Whereas if we're actually putting hardware there, we're not simulating so we can't predict. So the other reason that we're interested in getting the latency right down is this chap here. If you haven't read this book, get it, but give yourself a year to read it. I started a year ago, I'm still only three-quarters of way through because literally it's so dense with insight that you have to take your time and think about it. But the critical thing that his experiments have shown in others in that community is that any time you're holding something in your mind, you are burning psychological energy. Even if you are pushing a button and waiting two seconds to see what that button did, that's two seconds of extra psychological energy that you have burnt. And for our students that are holding down a full-time job, if they're coming into a remote lab virtually on a Friday night after an exhausting week at work, and every time I ask them to push a button, they have to remember what the button was for two seconds. I'm costing them psychological energy they probably don't have. And in the modern attention economy, we're calling it an information age. It's really an attention economy that we're in. There's so much information out there. It's can you get attention onto yours? If I don't like the start of a YouTube video, I've clicked off within four seconds. So I'm certainly not waiting for somebody's hardware to take four seconds to respond to something. And I don't think the students will accept that either.  You don't want three minutes or even a couple of seconds. So in one of the rooms in our lab, I have been beavering away trying to combine some existing technologies to make them do video quickly.  You speak, the other person speaks, and you're having a pretty natural conversation even all the way around the world. So that means the technology exists to get video responses from hardware really quickly. But it's a little bit trickier to set up than just a standard one-port streaming media server. So the way it works is that the two peers need to find a way to share some information with each other about where they are in the world and how they can be communicated with. So they need to talk to a third party and that third party has to figure out where they are and join them up. And once they've done that, they start trying to communicate with each other and if the connection is successful, they're sending information to each other directly without the third party being involved. So let's get rid of one of the people and that's not open hardware, but it is hardware. Let's put some hardware in there and have the hardware be one end of that video connection. And let's get that low latency from that peer-to-peer. So it turns out that works and this is the very, very first rubbish interface that I built. But let's say I finished this about two or three in the morning, phoned my dad and went, Dad, would you like to play with some real hardware in the UK? Of course, it was Sunday afternoon for him, so perfect timing, he was wide awake. I gave him a web link for this and within about four or five seconds, the pendulum on the other side of the room started going and swinging about. There's this little rotary pendulum thing here. So basically, within about four seconds of me saying to someone on the other side of the world, would you like to play with some hardware? They were interacting with it in real time. There was no software to download and the responses were good enough that my dad kept logging in every day for weeks, figuring out how to make it do cool stuff like swing really high and he was like, son, if you do this with the controls, it was really cool. And it turned out that this thing that I thought would be really quite straightforward and not have much depth to it became almost a telepresence device in my house for a month. So that was pretty cool. So that connection, let me see if I can turn that on. Could you click on the video for me? It's the bottom left of the click pad to make it click. Yeah. Oh, right. I know what I'm aiming for. No, no, not at all. That's trying to do this from the mirror. Yeah, here we go. Okay, so you'll see here that there's a green dial on the bottom and there's a little mouse on that. That's my dad moving there in New Zealand and that's what he sees on the screen as he does it. So you'll see him move it back around again. So that's the latency recorded on the screen from New Zealand with equipment in the UK. So I thought that was pretty cool and we've gone on to build a bit more. Can you pull the same trick again? This is a slightly improved interface. This was when I was in Austin presenting this at National Instruments Week a couple of weeks ago and I swear that's better than I've had from one side of Milton Keynes to another. You know, genuinely I'm not faking it that is video taken from my laptop in a coffee shop somewhere in the east side of Austin. Back to New Zealand. I mean back to the UK, I should say. So I've had people try this out all around the world and basically it works. So what I want to do is show you a little bit more about how this is done so that we can start moving toward making this sort of approach available to you for use with your projects. So the architectural philosophy for this actually comes from what Ed and his team have been doing already which is make things simple. Put everything in a JSON object and send JSON objects back and forwards between your experiment and your user. And that way you've got a nice cut out. You can separate out the concerns of how the equipment works and how the web app works. So my web apps know nothing about the internal details of how the hardware is run and the hardware server has no idea how the user is processing the data they get. It just says, here's your latest sample. You do what you like with it. If you want to store it, if you want to re-zoom in, if you want to zoom out on the graph, your problem, don't ask for the data again. You've had it. And that keeps things quite straightforward. This stuff works on a mobile phone as well, actually, but mobile phones don't do fast food. It transforms very fast, so there may be some exceptions there. Now, the critical thing where this starts to get slightly more involved is that as of December 2015 you cannot have a mixture of secure and insecure stuff on a web page. So if you want to do WebRTC, you have to have a secure web page. So that means you can no longer do insecure web sockets on the same page as WebRTC video. So that means that a lot of the stuff that's around for doing web sockets on open hardware, you quite often see a line that says, oh, it doesn't do this securely. So there's a small gotcha there, but I can tell you how to get round that. So the critical line was this one here where it says there's an extra S. So you've got to get that in there.  I mean, they're better than the video.  So, you know, as far as connecting to hardware goes for control and data, you could do that in WebRTC with its data connection, but Web sockets is, how should we say, it's a direct connection, so you don't have to go through something. I'll explain next, which is setting up the video call. So it's kind of nice when you go to a page to have something happening straight away. If you do do the data and control with Web sockets, that's a much more bombproof thing to set up than a WebRTC connection. So you can get data showing on the screen while the video is loading, and I think that provides a sort of a nicer user experience. So Web sockets are still in. Very good. Peer to peer. There's not really much point in investing a lot of effort in something if it's not going to be around for much longer and if there's not a very big community behind it. But WebRTC is in the right place at the moment. It's getting a lot of attention from the major browser companies, so there is a huge, huge, huge amount of investment going into making sure that the video is coming out of your webcam and getting onto the internet as fast as possible. So this is where the latency thing gets solved. Basically, WebRTC, you're not touching the video very much. It comes from the camera, it ends up on the internet going to the other person. That's why this is a preferable approach to others. It's just because it's going onto the wire and going straight there. Believe it or not, Microsoft's coming to play. ORTC, their version of RTC, is going to synchronize with WebRTC at standard version 2.0, or so we believe. So it means that it's the right kind of place to be investing in if you're doing development because it should be around for the foreseeable short-term, medium-term future. How'd you connect? I know this sounds trivial, but one of the things that started doing my head in when I was trying to get to grips with this at the start of doing it was how do these two peers actually choose one another amongst all of the other peers that want to talk to each other in the world? Why are you not accidentally ending up talking to somebody else? If I go to my server and I put in the room name that I want to talk to someone at, why is that room name not then able to be accidentally shared by somebody using a different server? It's actually really quite straightforward and that is that this line here, you can't really see the red background here, but this URL here is where your signal master is. That's that computer at the bottom of the screen that says, hey, I've got people that want to talk to each other. So if you go to a particular signal master, it's like going to a conference center, and then if you give it a room name, that's like going to a particular room in that building. So you have to get to the right building and then the right room in the building, and then the master with the right room name in order to speak to each other. And the thing that I'm doing to save a bit of bandwidth and keep the student's privacy is making sure that the student's not sending video back to the equipment, because quite frankly, the equipment does not care what you look like. A few of my friends got a fright when they realized that their video was actually showing back the equipment when I had the early prototypes. So I very quickly decided I needed to get the video coming back off. But it seemed wrong. So that's that bit there. So you've got full control over video and audio. At the moment, I have audio turned off for the system that's running in my house. I have it turned off for the systems running at work as well, but we're turning that on soon enough because hearing something clank and bang if it moves is kind of cool. It's part of it.  Yeah, we want to check that. And you can check because if you go to your own web page and you get a lot of feedback, audio, then you know that you've got your room bugged. So in our case, each experiment has its own room. So we have a signal master that we're running ourselves and each experiment has its own room name within that. And we'll keep changing them just to keep students on their toes. Then we get to the bit that, thankfully, there are a lot of smart engineers working on at the various browser companies, but it's that whole process of how you actually connect these peers up because, well, let's face it, if you've got your own broadband connection at home, it's very, very likely it is behind network address translation because there are not enough static IPs for the UK broadband providers to give you your own static IP address. So that means that the IP address that your computer has is probably going to be a 192.168 because it's behind network address translation behind your home hub router, which itself has an IP address which is behind network address translation. So you're more than likely to find that your computer doesn't actually know what IP address to give to the signal master, to give to the other person because if you each give each other 192.168 addresses, then you're not going anywhere. So what you have to do is ask each peer to talk to a server that can identify it. A stun server does that job. Basically, all it is is a computer that has two network cards so that some of the clever tricks that people play with pretending to have slightly different IP addresses depending on what IP address you give them can all be worked out. So that server at the bottom of the screen that I said does a connection actually has multiple parts. One of them is, yep, these are the two people that want to connect, and the other is this is where they're actually sitting at. And that stun server will generate a number of different candidates that could be connectable between each person. And if it all works, great. You might notice when you're using different WebRTC products that they all have a slightly different experience as you connect. People have different degrees of paranoia about how quickly they want to connect and how good they want that connection to be.  Some go with something that's really boring and slow and then upgrade if they can. And it all depends on the architectural philosophy. So the reason why peer-to-peer looks different in so many different products is because of these decisions that are made about how you hook up. Is it really that hard? It depends on your IP policies, your firewall policies. So it seems like an obvious thing to say, but if you're sitting at a castle, you want the things that are being thrown by catapults to go out. You don't want them to come in. And if you imagine that your router is a little bit like a castle wall, if you've thrown something out through it, there's temporarily a hole in the air that has gone out through. So something could come back in through that. And that's effectively what happens when you do a peer-to-peer connection. You have your firewall and the other person's firewall. And once you have each other's details, you just start spamming each other with binding requests. And what you're hoping is that your binding request went out creating a pinhole in your firewall where one of the messages that they're spamming out comes back in making your firewall think the other computer genuinely responded to you. But what actually happens is you're just both spamming each other at the same time, hoping that eventually holes open up on your own firewalls and the messages will get through. So this is why it's not as simple as a WebSocket connection. There's an element of jeopardy here. So if you use TShark or Wireshark, you can see an absolute shed load of binding requests go out. And if you don't see a success response like this popping up, they're easy to spot because they're actually more characters. So you can see a long block of text here, an extra line kicking out now and then. If you don't see the binding success response, no joy. If you don't want to TShark or Wireshark, you can check those connections in browsers. They'll go through a list of all the candidates that were tried out. But if it does work, then you can show your prototype off. And one of the things that I want to do that I'm working on at the moment is creating a light version of that demonstration that I showed you before with the pendulum that's built around using Open Hardware rather than relying on the National Instruments Kit that we've got. So a poor little Beaglebone Black, I have roundly abused by sticking all sorts of heavy servers on it, and it's basically looked at me and laughed and gone, that's easy. I can do that all day long. So I can run a full WebRTC stack on a Beaglebone Black, and it doesn't seem to be bothered particularly heavily by that. So the thing that it doesn't do very well is the video. So a Raspberry Pi and a Beaglebone Black, neither of them have particularly good frame rates with USB video cameras. So funnily enough, it's doing all the complicated stuff, but the thing that needs a GPU, it's struggling a bit. So actually, if you do try and do a single board solution to controlling something, running all of the WebRTC servers, and having the camera, the thing it falls over on is just the USB bandwidth and the processing of that camera stuff that comes in. So I think the X15 probably will solve that problem, because it looks like it's got laptop-style capabilities. So you'll have to spend a little bit more than 35 pounds to get one of those, but I'm hoping that that might provide a solution. There were some other gotchas on the way through, which is that the ARM version of Linux doesn't have a network on line targets, so it tells your system that the network's up and running when it's not. So you start your turn server or stun server and it says, yeah, I can't run right now, but that seems to have solved itself by taking the correct files from another distribution and dropping them in. And the other gotcha at the moment is I'm using Namecheap for my domain hosting and my SSL certs, because self-signed certificates aren't good enough for that what's the word, mixture of security levels solution. And they only offer a Windows client. So I have to have a Windows machine running somewhere in my house to keep Namecheap updated with the IP address I'm currently using. So it's just stuff like that that's preventing it from being a completely open hardware solution. All right. So what's next? Well, I've got an absolute shed load of work to do with the team back at Milton Keynes to make all of our hundreds of experiments come online. So over the course of the next year as in when time permits I'm planning to just keep putting a little bit of work into making this available in open source fashion. I've talked to everyone from the dean all the way up to the vice chancellor all think it's a wonderful idea to release this as open source. Now I have to convince our commercial department. So we'll see how that goes. It should be okay. And I guess I'm going to have to try and convince someone to find me an X-15 so I can try that one out. But the aim is that that should go swimmingly well the next year on a Sunday we can do a workshop on connecting your hardware up to anyone in the world using this approach so that you can do what I'm doing with my dad with your cool hardware and hopefully make some money on Kickstarter. That was basically this bit. Just do whatever you can to connect people to your stuff. Make it compelling. And what we're doing the thing we're showing off okay has lots of bits and pieces in it but to do this with one piece of hardware I think it's an entirely doable goal for you all. So partying thoughts, use video to connect your projects care about latency and in our case the thing we want to show off is our massively parallel remote laboratory facility for our electronics course and fun enough we actually have heaps of room for expansion to other courses so if there are things that you know about you think we might be interested in partnering on to do cool stuff in our lab do also let me know. Right there's a huge team of people, there's just a few of them that are helping out with this it's one of them and that brings me to the end of my prepared comments. Thank you very much.


The governor of Imo State, Hope Uzodima, has sucked all his special advisers and senior special assistants. Speaking to newsmen in Government House Inowari, the Commissioner for Information, Declan Emelumba, said their appointees were sucked following a recommendation of a performance review commission set up by the governor to assess the performance of appointees. And the report recommended that there is need to re-jig that aspect of governance. And so following that, the entire SSS and SSS as a body was dissolved today. And the governor will also announce that he will reconstitute it and looking at the reports will determine whether the number is necessary to cool down or otherwise. And maybe those that might also be considered official.


Welcome to this lecture on basic calculation on HST system and hydraulic fluids. This is a tutorial sort of lecture. Now, look into a typical question on basic calculation of HST system, which may be one of the five questions of equal credit to be answered in a three hour written examination. Now, in this question part A, a hydrostatic transmission system consists of a fixed displacement motor of capacity 25 milliliter per revolution and a variable displacement pump. In the pump and motor the maximum flow is 500 millilitres per second and the maximum circuit pressure is 70 bar. Determine one the maximum power in kilowatt to maximum speed of motor in rpm and maximum torque in Newton meter available at the motor output shaft assuming all efficiencies as 100 percent. This means there is no loss neither friction loss nor leakage loss. Now, this question has another part which is part B in that it is asked if a constant power output of 2 kilowatt is to be developed. Determine one the minimum speed at which this power can be developed and two the torque in Newton meter available when the motor is running at maximum speed. Part C if below the minimum speed obtained from above B the maximum torque is maintained constant. Determine the speed at which 20 percent of the systems maximum power output is being developed. Now, here I would like to mention that when you first read it many part may not be very clear, but let us see this how it is answered and then it will be clear to you. Now, here also the time may be 33 to 35 minutes to answer this question. Now, let us consider the part A answer to the part A. Now, what are given fixed displacement motor of? We have been given the flow revolution per minute that is basically geometric displacement. I have mentioned this machines are positive displacement. So, geometric displacement is 25 milliliter. What does it mean? When you rotate that pump by in this case no it is motor rotate by one revolution the shaft total displacement will be 25 milliliter which is termed as swept volume. Usually it is called swept volume. Now, one important factor is there. If we call the geometric displacement then we call say 25 milliliter per revolutions, but swept volume means it is always per revolution. So, we do not use that word per revolution here. Swept volume is never written as 25 milliliter per revolution rather we write 25 milliliter. This means that 25 milliliter per revolution is the geometric displacement. Variable displacement pump in the circuit that mean actuator instead of a piston there is a variable displacement pump sorry in the pump side there is a variable displacement and the actuator side there is a fixed displacement motor. Between the pump and motor the maximum flow Q is 500 milliliter per second and the circuit pressure maximum is 70 bar. The maximum circuit pressure may be considered as pressure at the output of the pump, pressure at the outlet of the motor minus the pressure at the outlet of the motor. Ideal case considering no loss in between. Now, what we have to determine the maximum power in kilowatt maximum speed of motor in rpm and maximum torque available motor with 100 percent all efficiencies. We assume it will take 15, 16 minutes for answering this part a. Now, first of all we should draw a circuit like this what is there this is the variable displacement pump and this is the fixed displacement motor. Now, obviously this is a pressure relief valve here the pressure is mentioned and here the flow rate maximum flow rate is mentioned. Now, one thing is there we are using this term bar, but normally nowadays we mention this pressure either Pascal or mega Pascal's, but you should know some term because very often people used to call the pressure is in bar in case of fluid power. We know that the power is given by flow into pressure. The power in case of which is transmitting torque how we define the power torque Newton meter per into omega that will be exactly the same dimension if you multiply with q into p pressure difference into flow rate. Now, you hear the flow rate in milliliter per second and pressure in bar which is kg per centimeter square. If we divide this by 10 that simply will give us what in case of SI units we can express this flow in meter cube per second and pressure in Pascal's and that will also give us what that means this is meter cube per second and this is Newton per meter square. So, this will become Newton meter per second. Now, substituting the values of pressure difference and the flow maximum power through pull is simply 500 milliliter per second into 70 bar divided by 10. If we convert this flow rate into the meter cube per second simply we can multiply with 500 into 10 to the power minus 6 milliliter means cubic centimeter and we are converting into meter cube per second whereas this 70 bar is we can simply put 7 mega Pascal into 10 to the power or 7 into 10 to the power 6 Pascal's. We should remember that 10 bar is equal to 1 mega Pascal 10 bar is equal to 1 mega Pascal 10 bar is equal to 10 kg per centimeter square that is almost equal to 1 mega Pascal's. So, this is expressed in SI units. So, therefore, power is 3500 watt and it is 3.5 kilowatt this system. The speed of the motor is given by simply flow divided by the swept volume. Now, here we are putting this swept volume but while we are writing this units we have put this revolution per regulations because that otherwise there will be unit mismatch. This means that when we substitute this value we must remember this is the unit of that value. Now, substituting this value which becomes 20 sorry here we have made a mistake this will be 25 this value is 25 not 20 25 because this is the swept volume is 20 by mistake we have written this. So, anyway this calculation the result is correct 20 rps which is multiplying with the 60 that is in seconds it will become 1200 rpm. So, that is the answer that what will be the speed at 500 milliliter per second flow with 25 is the displacement volume. Now, the torque to calculate the torque simply if you remember this this will be very helpful. What is the pressure drop and the swept volume but look at this units you have to put in this way. So, swept volume and the pressure divided by 20 into pi in case of this CGS system whereas, in SI units you simply put the pressure in Pascal's and the swept volume meter cube or revolution and divide by 2 pi that will give you the torque. Now, if you substitute this value then this becomes 27.85 to Newton meter. So, this is the maximum torque available from the system. Now, part b in part b what are given constant power output of 2 kilowatt is to be developed we have to develop 2 kilowatt power only this much is given. So, this confuses what should be the pressure then now immediately we should consider that pressure will be 70 bar and we have to calculate what will be the minimum speed. The torque in Newton meter available when the motor is running at maximum speed that also we have to find out. Now, here I have given this note required additional data to be taken from earlier data what we have taken from the earlier data as we have to calculate speed we assume that maximum pressure is 70 bar that we have to assume that we should not be confused we should assume this value the minimum flow at which 2 kilowatt is developed is then we are developing 2000 watt. So, Q mean into 70 bar by 10 or therefore, Q mean is equal to 2000 by 7 to 85 milliliter per second which is 2.857 into 10 to the power minus 4 meter cube per second. So, this is the answer minimum flow at 2 kilowatt with pressure 70 bar or 7 mega Pascal's the speed of the motor at this flow is that is the amount of flow milliliter per second divided by the swept volume you need not you may write the formula here or simply you can substitute this because we have already used this formula earlier in answering you may simply write this and you can get the answer is 11.43 revolution per second and if we convert this 685.68 rpm interestingly if you take the ratio because pressure is constant if you take the ratio of 1200 divided by 685.68 that will give the ratio of 3.5 by 2 you understand my point for the maximum 500 milliliter the power was 5 kilowatt at 70 bar and for this flow power is 2 kilowatt all other parameters are changed. So, definitely the this rpm ratio should be equal to the power ratio that you can compare. Now again the part question of part B assuming that the flow to be 500 milliliter per second at maximum speed we would calculate the pressure drop you should go back that question part B and 2 there we have been asked what would be the torque at maximum speed now when this is asked that maximum speed definitely we have to consider we are considering the maximum system flow which is 500 milliliter per second in earlier case we have considered maximum pressure to calculate the flow in this case we shall consider the maximum flow to calculate the power. Now this pressure drop here is 2000 the power into 500 and this 10 is coming in CGS systems which is 40 bar in case of SI system simply what divided by the flow in meter cube per second will give us the pressure in Pascal's and this is equal to 4 mega Pascal's simply you divide this what divided by the flow rate in meter cube per second that will give the pressure in Pascal's. So the torque output Tm2 at that situation is that 40 bar into 25 is the swept volume or displacement per revolution divided by 25 CGS system formula is like that or simply the pressure is 4 mega Pascal into this flow is milliliter per second divided by 2 pi if you put here in Pascal's put this as a meter cube per regulations but if you put into mega Pascal then you can simply put as a milliliter. Now this is one interesting so this results is 15.92 Newton meter that is the maximum flow minimum torque at 2 kilowatt power. Now you should remember this torque Newton meter mega Pascal into milliliter per second or Pascal into meter cube per second because to this from mega Pascal to make it Pascal we have to multiply with 10 to the power 6 from milliliter to meter cube we have to multiply with 10 to the power minus 6 it is better to remember this relation. Now for the part C the maximum torque is maintained constant at below the minimum speed obtained from above B then the speed at which 20 percent of the system's maximum power output is being developed. Now here again this is a 3.5 to help not 3.8 20 percent of 3.5 is 0.7 kilowatt so 700 watt if torque is maintained constant at 2 kilowatt you see we have consider that minimum power is 2 kilowatt that we are maintaining 685 rpm the pressure would be 70 watt which we have consider. Now then the flow at 0.7 kilowatt it will be 100 milliliter per second the simple conversion and that gives us 4 revolution per second which is 240 rpm. So there it concludes here of course I have taken about 25 minutes to discuss but while you are writing and answering it would take 30 to 40 minutes not more than that if you know the how to solve the problem. Now we shall discuss another problem this is also a typical questions on hydraulic fluid the full question is what essential properties a hydraulic fluid fluid in hydrostatic transmission systems should possess what are the factors to decide it is viscosity what is viscosity index and what is its relevance in this field what properties are taken care you sorry is taken care are taken care of by additives what are fire resistant fluids what are their limitations this means that limitations to fire resistant fluid. Now this is also about 35 minutes to answer now first of all we shall discuss what essential properties a hydraulic fluid in hydrostatic transmission system should possess. Number one is the viscosity all hydraulic systems have viscosity limitations in the form of maximum and minimum values and optimum value may be recommended for highest efficiency if you look into any system you will find that it efficiency is not same at all operating conditions this will vary. Therefore, to achieve a performance required performance usually we should look into that what should be the while temperature to achieve this performance in terms what should be the viscosity within that efficiency zone. What we should take the precaution or what measures we should take if the efficiency is not achieved in case we find that due to the change in viscosity while density is changing we either have to provide cooler in most of the cases in some times we have to provide the heater also particularly when it is being operated in cold places either in a cold country outside or may be in a warehouse where it is say in inside the cold storage where we are operating that in a cold environment. Typical hydraulic fluids have viscosity equivalent to 150 to 225 say volt universal seconds at 100 degree Fahrenheit that means 38 degree centigrade. Now this you have to write this statement because this viscosity is normally considered for mineral based while the essential properties of fluids which I have already started discussing the viscosity index next is the viscosity index. Now viscosity already we have mentioned now viscosity index although it may not be very helpful for all calculation purpose but that indicates the variation of viscosity with temperature this is a you can say a scale which denotes the variation of viscosity with temperature. This scale is from 0 to 100 higher the viscosity index greater the stability range with temperature variations. Now next important factor is the stiffness through oil is though oil is considered incompressible hydraulic oil is considered incompressible. However it has a definite compressibility normally mineral based hydraulic oils have a bulk modulus of 17.5 into 10 to the power 3 bar which is 1.75 into 10 to the power 9 Pascal's this bulk modulus is comparable to the modulus of elasticity in case of metal. Gases and air entrapped in oil essentially reduce the bulk modulus which reduces system rigidity. Next is anti foaming characteristics as oil is vigorously circulated it tends to form foams which affects system operation antioxidant over a long period of time the oil gets oxidized when oil get oxidized it give rise to sludge you understand sludge this is like a the mud being precipitated at the bottom of the in a pond. So, in tank in oil tank you will find that a thick layer is being developed at the bottom that is due to the oxidization of the fluid a good oil must have long life usually depending on the operations this oil should change in case of say for example, the aircraft operation may be only few weeks whereas, in case of general purpose hydraulic normal industry you may find the same oil is used being for years anti corrosion the oil is to protect all metallic parts coming in contact with it by forming a suitable coating you will find mostly we use the ferrous component and this sometimes get corroded in with contact of some component of the oils not only this part ferrous part other parts also. So, we have to take care should it be hose should it be pipe should it be components of the device the samples of oil which cannot protect metals are considered unsuitable not only metals may be any component is considered unsuitable. Now, next is anti wear oil is expected to form stable films between the metallic bodies in contact carrying load and effectively reduce wear this property of oil is most important in fluid power applications in most of the components the parts are moving. Now, if wear the load is being carried and there is a relative there is a surface in relative motions if the film is not generated there will be direct contact and wear. So, we should be careful not only about the oil property, but also about the motion, but definitely there is oil property is an important factor we should not select a oil which is not having this property fire resistance. Now, in this fire resistance in if we consider the mineral oil this are inflammable. So, we have to look into this factor inflammability factor. Now, it become dangerous in operation particularly if the fire point is low. So, we should look into the heat generation when the system is being operated. However, if we find say for example, a machine is being operated throughout the year it might be when in summer temperature is very high then the oil temperature also will rise to a flammable situation in that case we should use a cooler. However, if there is we can add some additives to make it fire resistance it is better. Now, non toxicity and absence of order the oil must be non toxic for the safety and personal and also free from unpleasant order. Now, lubricity is directly related to the anti wear availability of the lubricity lubricate moving parts. Now, it might be in some fluid with the hydro dynamic action that generation of flim is better, but it may not have much lubrication property. So, lubricity is it some property inherent property in the oil which gives even if not favorable condition the condition is not favorable that is there is no wage action and other things, but still it should provide the lubrication lubrication of the components. Now, what are the factors to decide its viscosity? The factors to decide the viscosity are type of pumps and motors. Some pumps are able to operate only with a close range of viscosity. Vane pumps are unable to operate with a viscous fluid just keep in keep this in mind when we will study the vane pump we will discuss, but vane pump is not suitable for viscous fluids. Gear pumps and piston pumps are on the other hand able to handle wider range of viscosity. However, for very thick oil we have to go for screw pumps. Say for example, if you try to pump grease using a pump definitely if you use a vane pump it will not be efficient at all. Maybe you can use a gear pump or even piston pump, but in that case screw pump will be the best speed of pumps and motors. It is easily seen that high speed machines are best operated with thin oil. As otherwise loss due to viscous drag may be high reducing system efficiency. This means that for high speed machines it is better to use thin oil, but using the very thin oil definitely this will increase the possibility of leakage. So, we have to make components are more accurate for very high speed machines, so that we can use thin oil. Pressure it difficult to operate a very high pressure system with a thin oil as leakage and consequently power loss will increase. What is viscosity index and what is its relevance in this field? Viscosity index as explained in section 1 b is a measure of stability of viscosity over a wide range of temperature. In all hydraulic systems there is invariably a temperature rise owing to power loss. If viscosity falls rapidly with the temperature rise the system may become inefficient as the viscosity may move away from the optimum value. A certain grade of oil is given viscosity index 0 and another is given viscosity index 100. Now I would like to mention as this question is asked what is viscosity index and what is its relevance in this field? Then not only this definition, but also you should show that how the viscosity index is determined and how it helps to choose a well. Now these two are we should remember that viscosity index 0 is for texas naphthonic oil whereas viscosity index is pencil vinyl paraffinic oil respectively. So, this means that pencil vinyl oil is very good from the viscosity point of view whereas this one is having very poor viscosity index on the other hand what we have already discussed this is good for antioxidant whereas this is not good. But now let us see viscosity index is estimated from the viscosity temperature characteristics of oils as illustrated in next slide. Now you can draw this graph and again you can in the you can draw this. In fact if we experimentally find the viscosity of oil and with temperature we will find for mineral oil at about 210 degree Fahrenheit all are having the same viscosity. On the other hand at low temperature particularly below 70 degree Fahrenheit or so which is not shown here this curve will not be achieved although I have drawn it this curve will not be achieved maybe this curve can be drawn from 70 to 210 degree Fahrenheit. Now usually viscosity index is given mentioned at 100 degree Fahrenheit and how it is calculate we take at that temperature say T is equal to 100 degree Fahrenheit we consider V0 is this one simply draw a straight line and take this V0 value and V value is also here of this point and Vi is here then the viscosity index is given by V0 minus V by V0 minus Vi whereas this viscosity index of the sample oil. Now as you see that the numerator is V0 by minus V that means this length directly and we are dividing by the total height this height not length this height V0 minus V is this height. So this height is larger means this oil is very close to this one and this is having higher viscosity index that means less change in viscosity with higher range of temperature. So it will be better for operations a good hydraulic oil should have viscosity index around 90 but it is not easily achieved. Now what properties are taken care of by additives in modern hydraulic oils additives are added to improve properties. Now when the fluid power was initially was being used the people used hydraulic oils that mineral based oil at that time there was not much additives so there was not much control over that properties of this fluid but gradually different additives was found out by research and these those are added to improve the properties of oil. Now first one is that viscosity index improver certain polymers and similar additives are added to improve the viscosity index. Viscosity indexes in excess of 100 are not uncommon remember that means the what the sample oil which we have drawn in between that may be below the viscosity index and almost horizontal in that curve. Antifoaming properties these acts on on the surface tension and prevent formation of bubbles antioxidants increase the oxidation stability of oil. Anti wear helps in the formation of stable oil films now this answer I have given very short form you may slightly elaborate particularly if this is a short note type question what properties are taken care by the additives then definitely these you have to additives sorry these you have to elaborate but I also mentioned at the same time it is very difficult to remember the names of different additives if you can remember you will have extricated for those. Now what are fire resistant fluids mineral oil is normally not considered fire resistant at these have definite flash points. I have discussed what is flash points because with the increase in temperature when the flame arises we should call the flash point for hazardous locations fire resistant fluids are often employed. Now for an example if you would like to operate in mines say coal mines where there are different type of gases which highly influenable and there if you do not have fire resistance oil that will cause to flame the gas inside the mine and very often we need to use the hydraulic equipment inside the mines. So we should take care of this factor and we should go for fire resistant fluids. Now if you look into this to make it fire resistant we can add some synthetic additives to the hydraulic oil mineral based oil that one is that synthetic oil mixture that means we can add some polymer with the mineral oils and we can make it fire resistant phosphate esters that is that itself a material which can be used as fluid chlorinated aromatics that is also can be used as fire resistant fluid. Now it might be water based water oil mix mixture is possible but water glycols is very often used for fire resistant fluid but definitely it may not have the very efficient performance as with mineral oils. Now water soluble oil emulsions this is oil and water mixture to make it fire resistant what are their limitations the fire resistance of chlorinated aromatics is very good this means for fire resistant fluid we can use this chlorinated aromatics whereas that of water based oil is fair some of these oils say phosphate esters and aromatics are costly. I have given you idea say for example the well good quality oil for servo mechanism of course it is not fire resistant the cost is may be 300 rupees per litre which is probably used in aerospace or aircraft 300 rupees per litre and in many cases we need more than 1000 litre oils for a aircraft but this fire resistant oil may not be that expensive but to have enlarged quantity this fire resistance oil it again becomes very expensive and life of these are not may not be very good that we should keep in mind. So with this I finish this tutorial part