Well, we are in a Galatians And we're getting into chapter 5 tonight. We're gonna be looking at Galatians chapter 5 verses 1 through 6 And the title of our sermon is stand fast and free in faith stand fast and free in faith And we're gonna walk through Galatians 5 1 through 6 and some Exhortations by Paul here Paul is winding up his argument and now he's gonna start applying his argument Directly to these believers in Galatia here who are in difficulty with the law And so let's start reading together in chapter 5 verse 1 The Bible says stand fast therefore in the liberty by which Christ has made us free and do not be entangled again with a Yoke of bondage Indeed I Paul say to you that if you become circumcised Christ will profit you nothing and I testify again to every man who becomes circumcised that he is a debtor To keep the whole law you have become estranged from Christ you who attempt to be justified by law You have fallen from grace for we through the spirit Eagerly wait for the hope of righteousness by faith for in Christ Jesus neither Circumcision nor uncircumcision Avails anything but faith working through love. Let's pray Now father in heaven Lord, we praise you and thank you Lord for the Salvation that comes as a free gift of your grace through faith in Christ or protect us from this damning error of legalism Falling into a trap a snare of the devil of looking at our own works our own performance as a basis on which we're Justified or righteous with God and please protect us from that father Just those fiery darts from the wicked one and please help us to see this clearly It is a subtle insidious sin and God we want to Be free completely from it to gouge it out of our lives So Lord where we air in this way God where we sin in this way Please Lord draw draw our attention to it convict us Lord by your spirit and Lord we pray that you'd grant us a repentance from that so that we can follow Christ wholeheartedly by faith Help us Lord where we spot this in our church to be loving and gentle about Correcting this we're proving this Lord by your word For the glory of your name for the salvation that we have in Christ If we seek to attain righteousness through works of law and the cross avails nothing and To us Lord called by your name your people the cross avails everything We praise you and thank you for that Lord. Thank you for our Lord and Savior Jesus Christ and the sacrifice For sins that he has so freely given us. We praise you Lord in Jesus name. Amen. All right Well, we are in Galatians chapter 5 1 through 6 and the the title of the sermon again is stand fast and free in faith and The theme here that Paul's been working out he has been Working through this letter to the Galatians Working out the point that Christian liberty Christian freedom this idea of liberty in Christ comes to us on the basis of faith in Christ in faith only all Men and this is an important Point that I think we've established if Paul is established and working through Galatians all men apart from Christ are in Bondage they are in slavery the only true freedom the only true freedom It's available period end of story is the freedom that we have through repentant faith in Christ Everything else is slavery is bondage and certainly this effort to attain righteousness through works of the law or attain right standing with God based on your performance That is bondage. It is slavery outside of Christ We're in bondage to sin Roman 6 verse 16 says do you not know that to whom you present yourself slaves to obey? You are that one's slaves whom you obey whether of sin leading to death or of obedience leading to righteousness We are in bondage apart from Christ to an accusing conscience In worse slavery yet if our conscience isn't accusing right We're in bondage to an accusing conscience outside of Christ We're in bondage are under the oppression of judgment apart from Christ Romans 2 verses 5 and 6 says But in accordance with your hardness and your impenitent heart You are treasuring up for yourself wrath in the day of wrath and Revelation of the righteous judgment of God who will render to each one according to his deeds That doesn't sound like freedom does it doesn't sound like Liberty Doesn't sound like joy in Christ. That sounds like oppressive bondage to judgment But we're also if you're apart from Christ and you're trying to escape God's wrath You're trying to escape judgment, then you're in bondage. You're in slavery under the law James 2 10 says for whosoever shall keep the whole law and yet stumble in one point He is guilty of all Romans 7 9 through 11 says I was alive once without the law But when the commandment came Sin revived and I died and the commandment which was to bring life I found to bring death for sin taking occasion by the commandment Deceived me and by it killed me. There is no freedom no liberty under the law And if you're outside of Christ, you're in bondage. You're in bondage to your sin You're in oppression under an accusing conscience. You're under oppression of judgment of God You're in bondage in slavery to the law if you're trying to establish your own righteousness to be saved and Under that law you're under the curse Galatians 3 10 says for as many as are of the works of the law are under the curse You're in bondage to the curse you're in slavery to the curse You're under the oppression of the curse of God if you're seeking to establish righteousness by works of the law Ford is written Cursed is everyone who does not continue in all things which are written in the book of the law to do them You see outside of Christ outside of the liberty that is available in Christ outside of the freedom that comes in Christ You're in bondage. You're in slavery. You're under oppression and it's amazing to me But sometimes genuine believers we have difficulty with understanding what this freedom what this liberty in Christ really is It's something that needs to be explained. What is freedom? What is freedom here? Paul is making the point that we need to stand fast in the freedom that we have in Christ Stand fast and free in the faith. So what is this freedom in Romans 8 verses 1 and 2? There is therefore now no condemnation. That's one aspect of freedom. You come to Christ. You're no longer under judgment You're no longer under condemnation There's therefore now no condemnation to those who are in Christ Jesus who do not walk according to the flesh But according to the spirit for the law of the spirit of life in Christ Jesus has made me free From the law of sin and death. So when you come to Christ, you're in Christ You're walking by the spirit Then you are free from the law and sin of sin and death and that is freedom When you born in sin in your depravity as children of Adam by your nature depraved Unable to please God unable unable to live for God unable to escape wrath on your own When you come to Christ and you are now free from sin and death Free from condemnation. That's freedom. That's freedom and that's the only freedom that is free That's the only freedom that is that's all we want, right? And to those that don't seek or understand the value of that freedom. They're blinded blinded by their own sin They suppress the truth in unrighteousness, and they're just treasuring up for themselves wrath in the day of wrath Thinking they're freed. They're thinking they're free because they're living their lives for themselves But we when we were outside of Christ Growned didn't we if you're a disciple of Christ before you came to Christ You groaned under the weight of your sin at some point you came to grips with your slavery You came to grips with your bondage You realize the veil was lifted your eyes were opened and you realize that you are in slavery You're under oppression that you are under judgment. You're under condemnation You groaned under that fearful expectation of judgment didn't you and that was one of the things that God used in opening Your eyes opening your heart to understand the gospel one of the things that God used to convict you of your condition Convict you of your sin so that you could be saved you came to grips with that bondage of your sin, but until we were set free from this bondage of this oppression the only freedom that we were able to have is Freedom in Christ that comes through repentant faith Wesley wrote to him, and I thought these words were Appropriate considering what we're talking about here this freedom that we have in Christ this bondage that we're in under our sin And he wrote come oh my guilty brethren come Groaning beneath your load of sin that sounds oppressive doesn't it sounds like you're in bondage and that you are You're outside of Christ.  You are cleansed in him You have all of that burden lifted you have the weight and oppression and Bondage of your sin taken away in Christ There's freedom in Christ, and that's the only freedom that counts is only found in Christ Otherwise it's bondage and oppression and slavery and here just shockingly And we'll see more as we walk through this passage why but here the Galatians want to abandon that freedom They have this freedom that comes only through Christ and they're turning their backs on it But we need to take a look at what this freedom is Defined in our understanding when we think about freedom in our context and our understanding Freedom the way we know it is basically doing whatever you want to do when you want to do it All right, that's just sort of defining what freedom is if I'm free I want to eat well I'm gonna eat what I want to eat and I'm gonna eat when I want to eat it right and I Partake of that freedom more than I should however That's that's free here if you're free to eat you're free to eat You can eat what you want to eat when you want to eat it no one's controlling your choices there You want to see a girl you you go see that girl and you see the girl You want to see and you go see her when you want to see her because you're free You have the ability to do whatever you want to do you think you're free and so on the second date you decide You want to marry that girl? You ask her to marry her you get married and now you're in slavery She doesn't turn out to be what you thought she was gonna turn out to be and you're back in bondage again The freedom that you think you have is not the freedom that you really have freedom in that respect is an illusion And you figure out married to this girl that freedom that you thought you had not cracked up what it not cracked up To be what it should have been The only true freedom that we have is the freedom that we have by faith in Christ that comes with being a child of the Kingdom being a child of his when you are Walking by faith in accordance with the spirit in accordance with your new nature that God gave you in Christ That is freedom and that produces joy love it produces the fruit of the spirit It produces patience and perseverance and that's what genuine freedom produces real freedom I want to give you an example of this and I remember freedom that we have in Christ is walking according to the spirit all right Freedom in accord with the spirit and in accord with the new nature Walking by faith in Christ living the Christian life in that way is true freedom You come to to Christ by repentant faith you turn from your sin put your trust in him living the Christian life You focus on him you keep your eyes on him you walk this life by faith in Christ right and you're doing that under the power under the control of the Holy Spirit at work in you because when you're saved God gives you a spirit and then you're doing that in accordance with the new nature that God has given you and In that there's great freedom the result of that is freedom. I think about it this way You got a business owner that wants to hire a guy to make sales calls, and he's got two options one guy Absolutely loves to make sales calls All right, and the owner wants them to make sales calls and wants them to make a lot of sales calls The other guy that he has the option of hiring hates to make sales calls doesn't want to do it So not a tough choice for the owner here The one who loves to make sales calls is free to make as many sales calls as he wants Right because he loves to make sales calls the owner hires him to do that job He's free to do that job in his job He experiences freedom in that because he loves out of his own nature loves to make sales calls The other guy is in bondage right if he hires the guy that hates to make sales calls That guy is going to feel enslaved in his new job. He's gonna feel in bondage under oppression Okay, keep that in mind. We're gonna work that out as we go forward here without faith in Christ Without walking the Christian life according to faith without trusting him without walking under the control of the spirit through faith There is no freedom Freedom even for the disciple of Christ comes not simply by a Forensic justification at the beginning when you were saved Freedom for the Christian is experienced by walking in accord with the spirit in accord with your new nature by faith in Christ and that freedom that experience of freedom in the Christian life is sustained by Faith in Christ as soon as you step away from faith You deny that freedom that is yours in Christ and you start putting yourself under bondage again And some of you know exactly what I'm talking about by way of experience. We'll talk about that more our freedom as Christians Is a product of our new nature in the same way that the one guy loves to make sales calls It's in accord with his nature to make sales calls He's going to love sales calls the guy that hates sales calls is going to be Burdened and enslaved and in bondage because he hates to make sales calls It's not in his nature to love sales calls. So he's going to be enslaved our Freedom that we have in Christ Jesus is a product of our new nature Right the liberty that we have in Christ is a product of the new heart that God gives us If you're if you don't have a new heart then for all of your effort for all of your trying for all of your Intensity and energy devoted to try to live the Christian life. You're simply going to be in bondage You're not going to be free as an unbeliever You simply are not going to have victory over sin You simply are not going to have the joy that comes from this liberty that's in Christ I simply aren't going to overcome and that is because you're not a believer. You don't have the new nature Freedom as Christians comes as a product of our new nature, and that's why in the Christian life that his Commands to us are not burdensome to the guy that's making the sales calls that loves to make the sales calls Are the instructions of his boss to make sales calls? Is that a burden to him? No, because it's a part of his new nature He loves to make sales calls to the guy that's Growning under the weight of sales calls when he reads his contract and he says, okay I got to make all these sales calls. That's a burden to him. There's no joy. There's no freedom in that Okay, that's why when you're a Christian and you're walking by faith you're walking in accord with the new nature His commands are not burdensome. Think of that girl All right, if you love the girl Does anyone have to tell you to go talk to her?  Sometimes that old lacquer that old finish has to be sanded off and we need to be reminded sometimes that we have freedom in Christ sometimes just like these Galatians we can slip into patterns of thinking or patterns of living that can Look like the old you look like the old man And that's why it's so important to be in a good church around brothers and sisters to encourage you to study Letters like the letter to the Galatians to help you with that To exhort you daily to stir you up to love and good works to remind you to keep your eyes focused on Christ and walk By the spirit not by the flesh because in that there's real Christian joy But if you're like me there have been times when you can slip into that Introspection we've been talking about or that morbidity that we've been talking about where the joy of your Christian life is just siphoned away And it's because we're not walking by faith. We're not walking in the liberty That is already hours in Christ that is hours by repentant faith in Christ and as soon as you start walking in That faith in that liberty then joy is gone victory is gone The work of the spirit sanctifying you is gone That ends and joy ends freedom ends and you're back in bondage again. So let me ask you Examine yourself right now. Are you burdened by his commands Is there something about your Christian life your service to him? you're walking the Christian life that is burdensome to you And you've got a couple of options there one is you've slipped back into trying to establish Right standing with God based on your own achievement or number two you're lost Because when you come to Christ You have freedom in Christ victory over sin power in the spirit joy love peace patience kindness gentleness self-control all as a fruit of The new life as a fruit of your new nature as a fruit of the liberty that is ours in accord with our new nature all by faith in Christ and so if something about your Christian service isn't Is a burden to you isn't generated out of love and affection for Christ Doesn't bring you joy in serving your Lord in that way Then it's something to look at either one you have slipped back into the same sin That these Galatians have slipped into that Paul is seriously rebuking them for or You're not a believer and you need to come turn to Christ and Called out to him for the freedom that comes only through faith in Christ But ours commands a burden you lack joy Do you have difficulty or are you having difficulty overcoming sin now to the genuine disciple of Christ? There is sin you will battle your whole life But the Bible promises Victory the Bible promises Progress the Bible promises an upward direction the Bible promises Observable progress Are you just wallowing in the mire? Have you returned your own vomit? Are you just wallowing in that or do you see that observable progress? Do you see that? substantial Direction of righteousness in your life as a result of the spirit or are you just wallowing in your sin? No victory over sin at all. Are you trying to be free and you aren't changing? If so, you may need to turn to Christ and ask him for deliverance That's the deliverance that comes through repentant faith in Christ When you're in Christ through repentant faith, there is liberty and there is freedom. There will be victory There will be progress albeit ultimate progress only comes when you leave this life when God takes you home But there will be progress not the perfection of your life the direction of your life the direction of your life is Observable change the direction of your life is Observable righteousness the direction of your life is a hatred for sin and a joy and a fulfillment in serving Christ a love for him a love for his commands So it's it's worthy in a situation like this Especially under the the rebuke that these Galatians are under here to examine yourself and to see am I in the same the same boat We meet we need true freedom and we need true freedom in Christ and here in Galatians 5 The Galatians want to give that up The thing that we want is liberty in Christ freedom in Christ and the Galatians want to give it up. Look at verse 1 Stand fast therefore. This is an imperative command from Paul here. Stand fast therefore in the liberty By which Christ has made us free and do not be entangled again with a yoke of bondage Now Christ's work is summarized here and it summarized in two ways one is through an Indicative statement we've talked about that before in Greek indicative is a statement of fact and the fact is is that you are free from the law in Christ if you are a disciple of Christ then you are free from the dominion of the law in Christ the imperative as a result of that indicative the imperative command is now Stand fast in that freedom. Don't drift from that. Don't turn away from it I don't fall this is paramount to the sin of apostasy here to turn away to turn away from that freedom To turn back to the law is tantamount to apostasy to turning away from the faith Okay, this is like the sales guy imagine Telling the sales guy now who absolutely loves to make Calls loves to make sales calls Imagine telling him now. Okay. I want to hire you into this job over here and in this job over here It's the same job. You got to make sales calls. All right, but you in this job over here. You get to make 600,000 sales calls a day Might as well be 600 billion 600 million 600 trillion. Okay over here. You just make sales calls over here We want to hire you into this role and you need to make 600,000 sales calls per day Or you keep your job over here and you're just free to make sales calls Okay, you see the analogy a little bit over here. You're free to make calls over here You make calls but you got to make 600,000 a day The law does that Under the law, there is no freedom. You have to perfectly Keep all of the law and it might as well be like that guy making six million sales calls a day, right? It's impossible under the law. There is no freedom to keep the law. There's only bondage. There's only slavery There's only oppression because you cannot keep it like the passage in James if you break one of the laws You're guilty of breaking all of the law But over here You're free and you're free to keep the law in Christ over here The law is a yoke in acts 15 turn to acts 15. Let's take a look at this The law is a yoke. It's a burden. It's oppressive. It's bondage. It's slavery and look at acts 15 We're gonna get here in a couple of weeks On Sunday morning But an acts 15 in verse 1 this is a synopsis of this Talking about the same situation here verse chapter 15 verse 1 and certain men came down from Judea and taught the brethren Unless you are circumcised according to the custom of Moses you cannot be saved Therefore when Paul and Barnabas had no small dissension and dispute with them They determined that Paul and Barnabas and certain others of them should go up to Jerusalem to the apostles and elders about this question So being sent on their way by the church they passed through Phoenicia Samaria describing the conversion of the Gentiles They caused cause great joy to all the brethren when they came to Jerusalem They were received by the church and the apostles the elders they reported all things that God had done with them But some of the sect of the Pharisees who believe rose up saying it is necessary to circumcise them and Command them to keep the law of Moses now This is the situation that they're dealing with here in the letter to the Galatians Circumcision is used as reflecting the entire law here. They're specifically in this passage specifically speaking about Circumcision, but that is reflective of if you keep certain the law regarding circumcision You've got to keep the entire law look at verse 6 now the apostles elders came together to consider this matter And when there had been much dispute Peter rose up and said to them men and brethren You know that a good while ago God chose among us that by my mouth the Gentiles should hear the word of the gospel and Believe so God who knows the heart acknowledged them by giving them the Holy Spirit Just as he did to us and made no distinction between us and them purifying their hearts by faith Speaking of uncircumcised Gentiles Verse 10 now therefore, why do you test God and here it is by putting a yoke of burden an oppressive bondage of slavery on the neck of the disciples which neither our Fathers nor we were able to bear But we believe that through the grace of the Lord Jesus Christ We shall be saved in the same manner as they this law is Bondage it's a yoke of slavery. It's taking the guy who loves to make sales calls and Putting him under the bondage of making six hundred thousand a day, right? See the difference there One sense he'd be free the other sense. He's in bondage Christ frees and Liberates the Christian from that yoke When you come to Christ by repentant faith, you are freed from that yoke of bondage. There's no longer the Impossible law to keep in order to be righteous before God now You are free in Christ to just simply out of a loving heart obey him And there is no longer the burden the slavery the oppression of trying to establish Perfection because that's what is demanded Perfection on the basis of your own effort in keeping the law That's simply bondage and Christ has set us free from that bondage Christ frees and liberates the law in Slaves and holds captive the Galatians here are foolish They're making a foolish mistake It will turn out to be fatal to them if they continue down this path and Paul here is going to give a scathing warning about that Freedom to the Galatians is already theirs if you're in Christ freedom is already yours You've already been removed out from under the yoke of that law You've already been removed out from under bondage out from under slavery now you are free Think about the guy making sales calls if you get the job you have the job where you just get to make sales calls Then you are free to make sales calls. Why would you ever want to? Take the job over here where it's required of you to make 600 million a day All right, and 300,000 of those between 9 and 9 30 in the morning.  You wouldn't do it freedom is already there's now want to make a distinction there that's not freedom to sin That's not freedom to sin like we talked about in the very beginning that true freedom is by faith in Christ under the control of the spirit Walking in accordance with our new nature in that circumstance you won't be slaves to sin The pattern of sin simply ends Because it's been broken by that reality Based on faith in Christ under the control of the spirit in accord with your new nature You're not going to be Wallowing in your own vomit over here. There's going to be victory over sin. It's not freedom to sin It's freedom to do what your new nature wants to do if you're in Christ Then your new nature hates sin and wants nothing to do with it If you're free in Christ by faith in Christ, then your new nature wants to obey him wants to please him and your Obedience your love for Christ your affection for Christ it all pours out of the new heart that God gave you, okay? the law simply Will not cure the desires of our heart that has no interest in the things of God If you are outside of Christ and all of this is just a burden to you It's all just hard work, you know human achievement trying to grit it out And you just don't have any victory over sin and your sin sin sin your attitude your the affections just aren't there for Christ The law will never cure the desires of that kind of heart what you need more than anything is a new heart What you need is a new heart in Christ And that is through faith in Christ and that new heart is free It's there's freedom. We need to stand fast in that freedom. Look at verse 2 Here the law in verse 1 is a yoke of bondage. Look at verse 2 indeed. I Behold I Paul this is serious business. There's about to be immense weight to this statement This is unvarnished truth. This is direct. This is pointed indeed I Paul say to you that if you become circumcised Christ will profit you nothing Bet says a commentary says that Paul here in this statement Mobilizes all of this apostolic authority to make this statement in verse 2. This is weighty. This is a this is not a matter of opinion This is not inconsequential. It's not a light thing. This is a weight This is serious business and we're gonna see their eternity is at stake here if they fall back and rely on circumcision they're falling back to the entirety of the law and That entirety of the law there is going to condemn them here circumcision circumcision relates to the whole law Look at Romans. Let's see an example of that Romans chapter 2 And look at verse 25 Romans chapter 2 verse 25 Here's an explanation of that from Romans For circumcision verse 25 is indeed profitable if you keep the law Do you see that? But if you are a breaker of the law your circumcision has become Uncircumcision therefore an uncircumcised man keeps the righteous requirements of the law will not his uncircumcision be counted as Circumcision whether your circumcised or uncircumcised Paul says matters nothing But if you are going to keep the law and if you're going to be circumcised then you're saying you're gonna keep the entire law verse 27 and will not the physically uncircumcised if he fulfills the law Judge you who even with your written code and circumcision are a transgressor of the law for he's not a Jew who is one outwardly Nor a circumcision that was his outward in the flesh But he is a Jew who is one inwardly and circumcision is that of the heart in the spirit Not in the letter whose praise is not for men, but from God Those who hold the circumcision and fail to observe the whole law Become like uncircumcised Gentiles if you hold the circumcision and Fail to observe the entire law and it's good for nothing you're like an uncircumcised Gentile You're gonna go to hell when you die That's the point that Paul's making here Paul teaches that no one can keep the whole law and Because no one can keep the whole law the circumcision profits nothing now profit there that word profit profits them nothing is related to Eschatological judgment it's related to last day's judgment in the sense that keeping it profits you nothing means Eternal torment if it profited you anything it would profit life, but it doesn't it profits you nothing if the Galatians here Submit to circumcision. We know what circumcision is Literally the word here is that they cut themselves off They cut themselves off from the only means by which they can be forgiven of their sins and that's through repentant faith in Christ Imagine the sales guy now The sales guy hired into the job in order to make six hundred thousand sales calls a day If he doesn't make the six hundred thousand sales calls in a day, he's fired I'm thinking about that for a second and relate the two he's got a job over here where he's free But he takes the job over here where he has to make six hundred thousand sales calls a day And if he fails to make all of those calls, and it might as well be six hundred trillion Then he's fired he loses his job when he was over here and free He had job security right but now he places himself under this yoke of bondage this yoke of slavery over here And if he doesn't do it if he doesn't get the job done, he's fired That means it profits you nothing for him to put himself over here profits him nothing He's going to be fired you get it. There is no hope for him to make those calls over here. He was free So why I mean why after being free over here? Would you ever? Come over here and do this. It just doesn't make any sense. It's not making any sense to Paul here either The point here is that the Galatians Believe somehow there's that there's profit in circumcision for salvation Somehow or another they've got their minds mixed up tangled up rats nest in there Thinking that there's some kind of profit in circumcision for salvation. Okay Then if they believe that now apply this to yourself and your own thinking if you believe that There's some kind of work. There's some kind of human effort human achievement whatever it is performance That profits you something then there is no saving benefit that comes from Christ The saving benefit that you had you thought you had from Christ is not there any longer You are cut off from the saving benefit that you thought you had in Christ You can't have both you can't have freedom and the yoke of the law you can't have faith and Works, they are mutually exclusive right and so back in Galatians chapter 5 you can only have one or the other and there is a Huge chasm between the two that cannot be crossed. This is mutually exclusive There's simply no way no way to do both back in Galatians 5 look now in verse 3 And I testify again to every man who becomes Circumcised that he is a debtor to keep the whole law. Okay, so let's think about our sales guy Over here. He was free, but now he takes this job to make his 600,000 600 million 600 trillion sales calls a day and They say to him well, I tell you what I mean, I know it's a lot It's a lot of calls, but if you make five calls between nine and nine thirty in the morning That's okay. We'll let you make five phone calls between nine and nine thirty in the morning But if you make those five phone calls between nine and nine thirty you got to do the 600 trillion But you can do the five between nine and I see the stupidity of that the same thing with the law Listen, you can come over here and you can do this one little thing of circumcision You can be circumcised or in Galatians 4 when they were keeping the Old Testament calendar days You can come over here and keep this portion of the law But what Paul is striving to bring to the forefront of their understanding is that if you think that way Sure, you can be circumcised, but you got to keep the whole law Sure, you can keep the Old Testament calendar days, but that's not where that ends You've got to keep the entire law or it profits you absolutely Nothing that here in verse 3 he uses the language and I testify again. It's like Paul in a courtroom It's got a courtroom feel here, and it's very serious. This is oath language.  This is covenant language here And we're talking about two different means of salvation one is perfect obedience to the entire law of God perfect righteousness or Christ the cross is finished work one or the other neither one you can't have both together They're mutually exclusive And it says in chapter 4 verse 21 before when we look at this that they actually desired they were Desiring to be under the yoke of this bondage. They Wanted it and so somehow in their minds they were Joyful about it. They desired that as if that were favorable to them and Paul here in verse 3 views that as great debt if you remember just by way of example or analogy the parable of the unforgiving servant the unforgiving man he had this Insurmountable 10,000 talents. I believe it was insurmountable I'm almost unimaginable debt that he was forgiven That he was forgiven in the same circumstance here the Galatians if they are outside of Christ then their debt is Unimaginable to keep the entire law that's outside of Christ the Galatians are professing to be in Christ and they Desire to go back under that debt and attempt to pay it It's just utter foolishness utter stupidity utter foolishness. This is the same thing when you are in Faith in Christ you're free in Christ and you attempt to return to slavery in legalism Slavery in works righteousness attempting to make right with God in your own performance It's the same kind of thing you're returning to this Unpayable debt and that's the way Paul is using it verse 3. He becomes a debtor to keep the whole law It's it's with that sales guys might it might as well be 600 gazillion phone calls Because 600,000 600 million. It's all the same there. It's an impossibility and Here with the Galatians. It's being made clear, too That their animal sacrifices are not going to help them any longer because Christ has come That sacrifice no longer works in the way that they thought it worked now There's Christ and so if you leave Faith in Christ and attempt to establish righteousness with God based on your own efforts You don't have the animal sacrifices to help you. It's all on you. It's all according to your own Performance they can't fail they can't even fail in one point and to fail in one point of the law is to lose it all Now you can't have the law and Christ and if they fail in the law, they've lost Christ if they make that transfer Christ is gone for them and verse 4. This is no veiled threat. Look at verse 4 You have become estranged for Christ that again that word there is severed cut off And we're gonna see this more and more here as we walk through Galatians They're severed from Christ cut off from Christ you who attempt to be justified by law You have fallen from grace if they try to be justified by law And they've abandoned the grace the Christ of the Gospels. They've abandoned Christ literally here This is a warning against apostasy. It's like Hebrews 10 Hebrews 6 This is so serious and this is why this needs to be so serious in your life and in my life as we walk the Christian life not to trail or Stumble into fall into the sin of legalism Because in departing faith it is so serious to fall into legalism and performance human effort human achievement that it is Tan amount to the sin of apostasy if you depart the faith for works You're departing Christ and you're departing the grace that comes only in Christ Law and grace are polar opposites law Tries to find righteousness in doing and it you just obey The righteousness the freedom that's in Christ finds righteousness only in Christ and he walks by faith And that's what the Christian life is these Galatians if they accept Circumcision if they accept the law in this way, they're abandoning Christ. They're abandoning The grace that is in Christ the Christ of the Gospels. They're abandoning the faith. That's why it's It's apostasy Now think about this way with the sales guy The sales guy who loves to make sales calls mind you Now is told by the owner of the business listen This job over here. It's not the one over here where he's got to make 600 million phone calls The sales guy making phone calls over here. He's got this job. He's free to make sales calls Now the owner tells him listen. This job is yours forever It's yours forever Make you're free to make sales calls make sales calls be free to make sales calls and you'll never lose your job You'll you have job security forever That's what we have in christ. That's a feeble. Forgive me.  It'll break down But think about it You out of your nature love christ You out of your new nature just want to obey him just want to please him You want sin out of your life.    You're just free to live for him You're free out of your new heart out of your new nature just to love christ just to serve him Just free and grace free in christ There's just joy. You've been saved your sins have been forgiven. You've been cleansed And you're just free to live and it's yours forever.   That's the liberty that we have in christ It's the freedom that is in christ and oftentimes even as a disciple of christ We get it's hard. It's sometimes difficult to keep that clear in our minds to understand that It's so easy to fall into the ditch of legalism Now this over here. You got to also avoid the ditch of license It's not freedom to sin And you need to be striving against sin You need to be gouging sin out of your life because out of your heart you want to serve the lord It's like paul and roman seven over here. You're saying wretched man that i am I want free from this body of death And you're struggling and striving against sin But you're free in christ Um think about this with your own life. How does this apply to you? If you're the disciple of christ that you know loves to make the sales calls And you've got the new nature that god has given you the new heart that god has given you Then are you or in what way are you? Trying to derive justification with god based on your performance And in what way have you and if you've been a christian for any length of time You're probably familiar with what that looks like how that feels what that smells like I mean you can see it coming because you've been there before But in what way do you attempt to derive justification with god based on your performance? You must trust only and completely in christ By faith in christ it simply cannot be done any other way look at verse five For through the spirit we eagerly wait for the hope of righteousness by faith When you're saved through repentant faith in christ You're made justified you're made righteous as part of conversion the the justification the imputed righteousness of christ Here what he's talking about is the final eschatological Last day declaration of god that you are righteous And that's the day that all genuine believers hope for there's great hope in the meantime while we're here We eagerly wait for that declaration We eagerly wait for that day and we walk by the spirit in faith trusting in christ And here between verse four and verse five There's a contrast that's hinted at in verse four you've become a strange cutoff from christ You who attempt to be justified by the law you've fallen from grace and then in verse five four We through the spirit eagerly wait for the hope of righteousness by faith. There's an implied or a hinted at Contrast there that basically states that true believers will not be Estranged from christ will not be cut off Nor will they defect from grace a genuine disciple is going to persevere till the end the grounds of that contrast are believers Rather than trusting in their own efforts trusting in their own obedience for righteousness Continuously Look at the grounds of their hope which is christ jesus By faith and the power of the spirit to live the christian life to be pleasing to them to obey It's the the disciple of christ is preserved and persevered in obedience Which is the fruit of faith Through the spirit's work through faith in christ and again in accord with your new nature It's the way it works. It's not the flesh It's not self effort. It's believing in christ believing in the hope eagerly awaiting the hope It's living for christ and this is again. It's a looking away from yourself If you're a disciple And you have this hope in you Then you're not looking at yourself. You're looking to your hope, which is christ You're looking at the cross. You're looking by faith at christ. You're living your life looking at christ It's not an inward focused. It's not a self focus. It's a christ focus That Looking at christ and living that way Is a product of the holy spirit. It's a product of your new nature. It's a product of regeneration It's a product of being a new creature a new creation in christ It's not produced by human willpower not produced by how much you try How much you bear it or grid it out is produced by the spirit And so if it's going to be produced in you You've got to walk by faith in christ for it Depending on him for it. So again, you know it putting An overarching understanding on what the christian life looks like If you are walking the christian life by faith Then everything comes through faith in christ. You need to overcome sin You want to get out of sin out of your life you turn to christ and by faith in christ You cry out to god for help in overcoming that you want to You know more faithfully serve in this area or you want to be stronger in this area or you have a need in this area It's all done by focusing on christ not on yourself. You want assurance of your salvation Assurance of your salvation is going to come through Perseverance over time by faith in christ and the the obedience the performance is going to be a fruit of that As soon as you take your eyes off of christ and you're no longer living by faith in christ Then performance and assurance go out the window because you're not walking by faith anymore You're walking by your own effort and so it's this looking at christ this looking to our hope this eager waiting this expectation All is an outward focus. It's all focused on what christ has done for you What is available to you in the cross What our final and eventual hope is the final and eventual declaration of our righteousness in that day All of that is an outward focus. It's just not it's simply not Focused on you at all and in verse six here. We're gonna see that that ends means everything look at verse six For in christ jesus neither circumcision nor uncircumcision avails anything But faith working through love means everything All right, that's what's implied there at the end of that verse Neither circumcision or uncircumcision avails anything now Less someone like these galatians or someone else take pride in their uncircumcision Paul attacks that on together to say listen It doesn't matter if your circumcised or uncircumcised none of that means anything But faith working through love It means everything but faith working through love That's the substance and faith here working through love some take that to mean that faith is the fruit of love Like you've got to be loving and that out of being loving faith comes completely It's a catholic in a wrong way of looking at that completely Here love is the fruit of faith and there's a couple of places where We make sure that that's understood clearly in Galatians five Love is a fruit of the spirit walking in faith is a fruit of the spirit Love is a fruit of the spirit love here is a fruit of faith. It's a fruit of the new creation On first Corinthians 7 19. This is interesting if we compare the two in first Corinthians 7 19 It says circumcision is nothing and uncircumcision is nothing do you see the comparison there? But keeping the commandments of God is what matters Are those two things contradictory? Uncircumcision is nothing circumcision is nothing but keeping the commandments of God is what matters here Paul's trying to establish that It's faith that matters And first Corinthians 7 19 those things harmonized perfectly well together keeping the commandments is a fruit of faith It's faith alone that saves but as the reformer said it's not the faith that is alone Faith without works is dead So faith alone saves but it's not the faith that is alone Here circumcision doesn't mean anything uncircumcision doesn't mean a hill of beans what matters is finding righteousness in him And if you are a disciple you hunger and you thirst for righteousness and in hungering and thirsting for righteousness Are you going to come over here and take the deal? Where you bear and grit that out in your own effort in your own energy having to keep all of it? Or you step over here in the freedom and the liberty that you have in christ And paul is telling you here with all seriousness mustering up all his apostolic apostolic authority to tell you Take this path This leads to death and damnation and apostasy This leads to life eternal in christ take this path and by faith in christ Seek the righteousness you hunger and thirst for righteousness Hunger and thirst for righteousness as you should and do that by faith in christ Not in your own power not according to your own efforts Now here it's interesting this whole connection with apostasy If the galatians continue down this path They go apostate They never were saved to begin with and they go to hell when they die if you if i persist in a a morbid introspective self-performing effort To maintain or attain right standing with god based on our performance And we just fall into that legalistic trap of i gotta do this and i gotta do that and i gotta do this And i gotta do that and you're doing that in your own effort Then the warnings of hebrus 6 hebrus 10 are for you The warnings here that paul is giving these galatians are for you Um, he's warning them to stay away from that path This is the grace of god in this warning the warnings of hebrus 6 hebrus 10 are a grace of god Those warnings in scripture are how god helps the believer persevere to the end those warnings are to Rouse up in you fear and they have i remember the first time i came across those passages scripture I mean it i had knee-knocking fear over those passages. Is this applied to me? Is this you know they can create and that's exactly the intention Um, they help they assist the believer stay on the right path here paul's admonition here his warning to these galatians Is the same thing it should cause you to fear legalism to fear Taking your eyes off christ everything that you have everything that you hope for is wrapped up in christ And so if one for one second you think somehow that you've got to grid it out in your own effort You need to cling to the cross that much tighter Turn to christ that much tighter. This is a warning fall away from christ and be damned And we need to keep clinging to christ keep trusting in christ Um, this should warn us to turn from this morbidity that's associated with our performance Uh, and turn from obsessive perfectionism If that's difficulty that you have or something you get tripped up on It's just the opposite you turn away from performance And you turn yourself to the cross you turn yourself to christ i read Luther on this passage and i just i love what luther said I'll read this to you. It just sums this up really well And see if this applies to you if you've ever experienced this before Luther said this is a very important and pleasant comfort with which to bring wonderful encouragement To minds afflicted and disturbed with a sense of sin And afraid of every flaming dart of the devil Think about that If you've ever been afflicted and disturbed Inwardly focused introspective morbid over a sense of your own performance A sense of sin and afraid of every flaming dart of the devil Then this these words from paul here should be a very important and pleasant comfort With which to bring wonderful encouragement to your mind Right he goes on to say Your righteousness if you're in christ your righteousness is not visible And it is not conscious But it is hoped for as something to be revealed In due time It's a glorious promise we have in christ Therefore you must not judge on the basis of your consciousness of sin which terrifies and troubles you But on the basis of the promise and teaching of faith By which christ is promised to you as your perfect and eternal righteousness The only perfect and sure and true righteousness that we can ever have is in christ And so when you as a disciple Find yourself torn up and wrecked over some circumstance What's the resolution to that you look by faith to christ and christ is promised To you as your perfect and eternal righteousness the only righteousness that you can have The only righteousness to you which is available at all And it's the righteousness that you have to have to be right with god Only available in christ and in christ alone By faith you try to secure it any other way The end of that road is certain death and certain damnation a fearful expectation of judgment So that's why it's so important and this is paul's warnings here in glacians 5 So critical and so weighty because that's it's such a serious sin You walk the christian life. We need to walk the christian life by faith in christ who is our righteousness. Let's pray Father in heaven lord, thank you lord for christ Just so lord in awe and that amazing gift Well, we're so unrighteous in and of ourselves god And you're so merciful and so gracious lord that In christ we have the perfect righteousness that you demand God help us keep our eyes focused glued on christ Help us lord to just embrace and cling to the cross And praise you god that you've given us the righteousness That we have in christ or that we have the hope of that One day final declaration on that day of righteous In him we praise you god and thank you for that hope god. Thank you for that promise And we know from your word lord that those our promises are just as sure as if they were already In the past They praise you and thank you lord for this salvation Now to keep us focused on christ Keep us faithful to him Keep our eyes glued on him. We're not trying to Reestablish or trying to attain some kind of righteousness on our own just something in our flesh lord that wants to Wickedly bend that direction, but lord protect us from legalism protect us from That wicked sin and just keep our eyes god focused on christ Would help us by your spirit walk this christian life By faith god and attaining the promises that we have in your word We praise you and thank you for them for your glory god and for your eternal praise and worship We pray all these things in jesus name


So good morning all of you so today we will look at in fact continuation of the last class which was extending great solution the great solution was originally proposed by grades for slug flow profile and it looks at the thermally developing region or the thermal entry region wherein you are hydrodynamically fully developed and you are looking at only the entrance region where the thermal boundary layer is developing for such kind of a that is what we call as region 2. So grades assumed a slug flow so where anyway the slug flow does not vary and it is a uniform everywhere and here developed solution for constant wall temperature which we had seen earlier so there the Eigen functions where what were the Eigen functions essential in the original grades problem so the Eigen the Eigen value problem there was actually a Bessel equation right so now the same problem can be extended to a case which is more realistic that is for a parabolic velocity profile okay it was this extension was done by a group of people cellars at all cellars tribe us and I have posted that in the moodle you can just have a look originally 1954 how they did the extension to the grades problem of course nowadays the solution is more by numerical methods they try to do approximate technique where they did something for our close to the wall the R which is somewhere in the middle and R which is far away from the wall and they patched up all the solutions together so what we are doing nowadays is directly go for a numerical solution to the Eigen value problem so the easier way to start is to introduce these non-dimensional variables for temperature as usual we define ? as T – T wall by Ti – T wall since this is a constant wall temperature boundary condition and your inlet temperature is also assumed constant so this is an appropriate definition for ? once you have a constant wall flux boundary condition then you cannot define ? this way okay because your when you take the differential with respect to X your wall temperature will not be a constant and therefore they will not cancel out on both the sides and this is the non-dimensional radial coordinate and your axial non-dimensional axial coordinate can be non-dimensionalized in this manner which is somewhat similar to an inverse of the grades number your grades number 1 by grades number is actually X by D by Peclet number okay so this is somewhat similar exactly now we are not using X by D here but X by R not so when we substitute into the energy equation okay the energy equation for thermally developing flow is this and if it had been fully developed your D ? by DX would have been 0 but in this case your flow is still developing and therefore if you substitute you get a partial differential equation in terms of ? which is a function of both the axial coordinate ? and your non-dimensional radial coordinate ? and the boundary conditions are at the entry region that is at ? equal to 0 your T equal to Ti so therefore your ? will be 1 and at the location R equal to R not that is at the wall okay so your coordinate system that you are looking is something like this this is your R and this is your X okay so this is your R not so at R equal to R not that is your ? equal to 1 that is where your ? equal to 0 where your T is equal to T wall so this is where you apply constant wall temperature and at ? equal to 0 that is at R equal to 0 center line there is a symmetry in the profile so therefore your gradient at the center line should be 0 so now once you assume that you can use separation of variables to solve this we introduce ? and we break up the solution as a product of two independent solutions one is a function of only ? the other is a function of ? so we introduce X as a function of ? and R as a function of ? and then substitute into the PDE and then we get to we reduce the PDE to two ordinary differential equations through the eigenvalue ? square okay and the cell the eigenvalue problem here is basically the one with homogeneous boundary conditions which is basically in the ? direction and if you compare this to the Bessel equation you can find that this last term here is not same so therefore this is not a Bessel equation if you look at these first two terms they appear similar but the third term is not the same as it is a function of you have ? if you multiply by ? square will have ? square into 1- ? square that comes different from the Bessel equation and these kind of general eigenvalue problems are called Sturm-Lüweil eigenvalue problems okay so any kind of an eigenvalue problem whether this is a Bessel equation or a basic ODE can be represented as a Sturm-Lüweil eigenvalue problem so the the Sturm-Lüweil eigenvalue problem can be cast into a differential equation like this of course you can express this also in this particular form and the Sturm-Lüweil problems have a particular property that when you integrate your eigenfunction you multiply your eigenfunction and you multiply it with the weighing function the weighing function is this which appears on the right hand side so this is the particular property this is your chronicle ? so if m equal to n then this will be equal to 1 otherwise it will be 0 okay so now if you compare your eigenvalue problem to the Sturm-Lüweil problem okay so you can write your eigenvalue problem into the Sturm-Lüweil form so that will come out as – D by d ? x ? dr by d ? plus the term corresponding to Q of x is 0 here and this term can go to the right hand side and that can be expressed as ? square x 1 – ? square x ? x r so I am multiplying throughout by ? and the first two terms I can write and combine and write it in this particular format okay now the format is the same as the Sturm-Lüweil format you can compare the coefficients your P of x is nothing but ? Q of x is 0 and your weighing function is 1 – ? square x ? so this is your weighing function as a function of ? okay so this can be written as 0 to 1 so this is strictly speaking a to b okay in your case your ? goes from 0 to 1 that is why I have just used 0 to 1 there so this is 0 to 1 and this will be R n of ? x r m of ? x weighing function here which is ? x 1 – ? d ? is equal to ? m n times 1 by 2 ? because here your ? square is your ? so this is square root of ? square which is ? n x dr n by d ? n x dr n by d ? at ? equal to 1 okay so this is this is how the corresponding property of Sturm-Lüweil comes out to be in this case and if you want to write the final solution for ? which is basically x into R so you know the eigen functions now in this particular case this is an OD and to get the eigen function you have no other option but to solve it numerical okay you can once again go back to your shooting method you can reduce it to do first order OD is and then here guess the value of ? because you do not know the value of ? unless you know the value of ? you cannot find out the eigen functions so guess the value of ? and once again you have to satisfy the other boundary condition and that is iteratively found out and that is the suitable value of ? so like this for the given value of ? you have a particular value of eigen function so for each value of ? you have to find the eigen functions and finally the solution will be a superposition of all these eigen functions okay so that can be written as a constant times the eigen function into the other solution for X the other for solution for X as a function of ? that is a very straight forward OD which can be directly integrated and that will be in terms of constant times e power minus ? square ? okay so this is your final solution for ? having determined your eigen functions and your eigen values you can therefore plug in into this expression and find the variation of the temperature I will just give you a representation of how the eigen functions look if you solve for them and if you plot the first three eigen functions RN 0 it varies between negative minus 1 to 1 0.5 so this is your R0 R1 and R2 this is a this is a representation of so these are not R1 R2 so here you have plotted with respect to ? here okay so this the so this is the eigen function corresponding to n equal to 0 that is for ? 0 okay and this is the eigen function corresponding to ? 1 so for different each eigen eigen value sorry for eigen value ? 0 ? 1 ? 2 you substitute into the OD and you can get the corresponding eigen function variation with respect to ? so these are the first three eigen functions so these are the most important ones there are other higher order eigen functions but their contribution will be relatively smaller so when you sum them you take only the first three or four important eigen functions into account okay so now the thing is this is the solution but still we have to find the constant CN okay so for this we have to apply the initial condition that is at ? equal to 0 and we make use of the property of the Sturm Louis when you integrate with the weighing function so this is basically since this is an orthogonal all Sturm Louis problems a lot orthogonal so this is a property which satisfies the orthogonal condition so this is an orthogonality condition of Sturm Louis system of problems and we will make use of this in calculating the constant so let me call this as equation number one and I am going to multiply both sides by RM and integrate so so for first before doing that I will apply the condition at ? equal to 0 which is equal to 1 therefore 1 equal to summation n equal to 0 to 8 CN you have Rn of ? into e power – ? n square 0 that is 1 so I can now multiply both sides by RM of ? and the weighing function okay so the weighing function is here ? x 1 – ? square d ? and integrate from 0 to 1 so this is also the same I have RM ? x ? x 1 – ? square d okay so multiply by this RM ? x the weighing function and integrate both sides so now I make use of my orthogonality property so therefore only for m equal to n this will be nonzero okay so this will turn out to be 0 to 1 Rn of ? x ? x 1 – ? square d ? on this side you can sum only if m equal to n so therefore this will be 0 to 1 and the constant can come out okay so this will be Rn square of ? because if m equal to n then only this will be 1 so this will be Rn square ? x ? x 1 – ? square d ? okay so that is therefore my constant CN will be 0 to 1 Rn ? ? x 1 – ? square d ? divided by 0 to 1 integral Rn square ? x so now I have to evaluate these integrals how do I do that for example the integral in the denominator how do I calculate integral 0 to 1 Rn square ? x 1 – ? square d ? yes so that will be 1 by 2 ? n x this particular thing right so I already know from the property of storm Louis will system of equations that the denominator 0 to 1 Rn square x ? x ? x 1 – ? square d ? should be equal to 1 by 2 ? n x dr n by d ? n x dr n by d ? at ? equal to 1 okay so this is what I get from integrating the denominator and how about the numerator how about the numerator you have a very nice clue you have basically the eigenvalue problem here from this if you integrate both the sites okay so if you integrate this that is basically you have already R x ? x 1 – ? square d ? and this will be therefore on this side minus 1 by ? square x you have ? x dr by d ? and you are integrating between 0 and 1 so at 0 ? will be 0 so therefore this should be at ? equal to 1 this will be 1 right so this comes from the eigenvalue problem itself so I can just integrate and I can find out the value of this okay so therefore now I can write my constant CN so if I substitute for this integral in this integral what will be the constant – 2 by ? n and so this will be dr by d ? at ? equal to 1 and denominator dr by d ? these two will cancel off so you have only this in the denominator so you have into 1 by dr n by d ? n corresponding to ? equal to 1 okay so this and this cancels off you have – 2 by ? n and this in the denominator so now for calculating this constant see as a function of n this is a function of n so for different values of eigenvalue ? 0 ? 1 ? 2 I need to know what is the derivative of the eigenfunction with respect to ? okay corresponding to ? equal to 1 so therefore now for different values of ? for ? equal to 1 I should know what is the value of the eigenfunction and then I should fit some approximate curve and calculate the slope okay so that basically corresponding to ? equal to 1 from there I can calculate my constant C okay so I can finally substitute for C into my equation number one for the solution so therefore the solution for ? can be written as – 2 this can be taken out and I have summation n equal to 0 to infinity and for CN so the other these are all functions of n so I have to just keep it inside the summation I have e power – ? n square ? into Rn of ? divided by ? n into dr n by d ? n at ? equal to 1 okay so I have simply substituted for C or C of n from what I have obtained here all right so finally one so once I know my eigenfunctions my eigenvalues and this derivative I can now finally find the solution for ? so this is my final solution so I will go further and I will calculate the expression for the nusselt number the local nusselt number so therefore now I will define my local heat transfer coefficient as K dT by dr at R equal to R0 divided by T wall – T mean this is my definition of local H and from substituting the temperatures in terms of ? and everything in terms of non-dimensional radial coordinates ? okay so this will be if you substitute for T in terms of ? you remember that ? is T – Ti by T – T wall by Ti – T wall okay so this will be Ti – T wall into d ? by dr at R equal to R0 divided by T wall – Tm okay now I can define a mean temperature non-dimensional mean temperature which is ? m as Tm – T wall by Ti – T wall so here T is a function of both R and X here T mean will be a function of only X okay so I can define a non-dimensional mean temperature and you see that T wall – Tm by Ti – T wall is nothing but – ? m okay so I can write this as – K d ? by dr at R equal to R0 by ? m what I can also do is replace R in terms of ? so therefore this will be ? into R0 and this will be at ? equal to 1 because your ? is R by R0 so I can replace directly with respect to ? so all I need to know is my mean temperature ? m and also the non-dimensional gradient of temperature at the wall so once these two are calculated I can find the expression for Hx so from the equation for ? let me call this is equation number 2 now I can go on and calculate the expression for first ? m so therefore I can write my ? m as follows so how do I define my bulk mean temperature according to the basic definition I take my non-dimensional temperature multiply it by the velocity and integrate across the cross sectional area so that is 2 ? into R dr okay so this is from 0 to R0 so this I divide it again by the mass flow rate so that is 2 ? integral 0 to R0 into u of R into R dr okay this in this cancels now I define my bulk mean velocity or mean velocity as once again 1 by so this is 0 to R0 u of R into R dr so this will be 1 2 this will be 2 by R0 square correct okay so I can now substitute for this right here in terms of the mean velocity so this is basically integral 0 to R0 ? into u of R into R dr divided by this will be R0 square into um by 2 okay so since um is only a function of X I can take this inside the integral and I can write this as u by um into R dr so I can now convert this completely in terms of non-dimensional coordinates okay so this will be 2 0 to 1 ? into u by um into ? into d ? because I have R by R0 here dr by R0 okay so this will be my expression for ? m as a function of non-dimensional ? and u by um okay so I can now substitute I already have my expression for u by um what is the expression for u by um that is from the fully developed parabolic velocity profile right that is twice 1- ? square in terms of the non-dimensional coordinate okay so I can substitute for ? coming from equation 2 and the velocity profile from this into ? m so so this will become – 2 summation of n equal to 0 to 8 e power – ? n square ? divided by ? n into dr n by d ? n corresponding to ? equal to 1 into 0 to 1 then I have so 2 into 2 4 okay into ? into 1 – ? square into R which is also a function of ? okay rn of ? into d ? just erase okay so I am just substituting for ? and u by um into this and I am grouping all the terms which are function of ? and that I am integrating from 0 to 1 so that is basically 4 times ? into 1 – ? square into R into d ? right so this ? into 1 – ? square rn ? d ? what is this value so we have already seen that this is nothing but 1 by ? n square into dr by by d ? at ? equal to 1 okay so we will just substitute that and my ? m now becomes so 4 into 2 becomes 8 here this is a – sign here okay just take note of it so – and – become plus here so 8 into ? n equal to 0 to 8 into so that is a dr by dr by d ? n at ? equal to 1 no this should be with respect to ? sorry therefore this comes out with respect to ? please correct it okay so this is dr n by d ? here okay so this this is one of the terms multiplied by e power – ? n square into ? and divided by you have ? n square here and there is a ? n here so this becomes ? n cube into already you have a term dr n by d ? n so this becomes dr n by d ? n corresponding to ? equal to 1 okay so this is your expression for ? m so now you have a closed form expression for ? m as a function of the eigen value and the derivative of eigen function both derivative as a with respect to ? as well as with respect to ? okay so now we need to still find d ? by d ? with respect to ? equal to 1 what is this value directly you can differentiate that is – 2 summation n equal to 0 to 8 e power – ? n square ? into this will be your dr n by d ? at ? equal to 1 divided by ? n into dr n by d ? n at ? equal to 1 okay so therefore we will substitute for ? m and d ? by d ? so therefore if I substitute I can so I can also write my so I have my expression for H x as – k by r 0 I can directly get an expression for n u which is defined as H into r 0 by k so it should be actually defined with respect to D therefore I will write this as H into D 0 and so this will be D 0 by 2 therefore this will be – 2 into d ? by d ? ? equal to 1 divided by ? m okay so I can substitute for D ? by d ? and my ? m so that gives my n ux as summation n equal to 0 to 8 a n e power – ? n square ? divided by twice n equal to 0 to 8 a n by ? n square e power – ? n square zeta where a n is nothing but dr n by d ? at ? equal to 1 divided by ? n into dr n by okay so I will give you a couple of minutes you can substitute and check for yourself it is just straightforward I am just grouping this entire term dr by d ? divided by ? into dr by d ? this has a constant which is a function of n okay so I call this as a n okay so this will be nothing but therefore a n into summation of n equal to 0 to 8 power – ? n square that is the numerator divided by the ? m also as the same term so in addition I have ? n cube here so therefore this will be a n by ? n square into this okay the numerator has basically 2 into 24 denominator as 8 therefore there is 1 by 2 there okay I hope all of you are clear I am going pretty slow here so therefore this is your final expression now to calculate my nusselt number here I need to basically know numerically the values of all these slopes of eigen function with respect to ? as well as with respect to ? and also the eigen values so from there I can calculate my nusselt number once I know the location axial location where I need to so my nusselt number is now only a function of my axial location non-dimensional axial location so what cellars did if you look at that particular paper which I posted so he did it numerically and he has predetermined the values of all these constants and tabulated them so I am just going to give you only the final values tabulated by cellars okay so you can also see the for the value of corresponding to n ? n the value of CN and AN okay so CN is required where in your solution for ? okay so that you also everything has been computed numerical so for n equal to 0 the most important eigen value that is your 2.7043 the value of CN is 1.466 0.748 and similarly for 1 2 3 I will also give the 4th value this is 6.6790 10.67 14.67 and 18.66-0.802 0.587 is 0.474 plus 0.404 0.544 so 0.462 0.38 so the most important the first five eigen values the corresponding constant CN and AN they were all numerically calculated and tabulated by cellars at all 1954 so therefore you can just directly use these values you do not have to sum them to so many terms you can just sum them to the first four or five terms okay you can directly substitute the corresponding value of ? A and in the solution for temperature the value of C then you can get an expression directly in terms of ? okay so which is basically your non-dimensional axial coordinates so for different values of ? you can actually plot and see how the Nusselt number varies right from the location where your thermal entry length starts okay now for the limiting case if you look at a very far away distance axially so that is for large values of ? for ? very large you can see that this is an exponentially decaying function okay so except the smallest value of ? if you go for larger values of ? that is corresponding to N equal to 1 2 3 4 so these are large values and it is already an exponentially decaying function so they will be very small okay so only the first term corresponding N equal to 0 will be important then you look at large values of ? so for that this will be reducing to your a 0 into so basically you have e power minus ? 0 square into something which is very large okay divided by two times you have a 0 by ? 0 square into e power minus ? 0 square ? right while I am retaining only the first term and neglecting all the higher term so these cancel out and this will become so your Nusselt number corresponding to large values of ? will be nothing but 2 ? 0 square divided by 2 so if you substitute the value of ? 0 corresponding to N equal to 0 what will be the value of NU so that is basically 2.7 the whole square divided by any of you anyone who is having a calculator can quickly check that 3.6566 anybody remembers the significance of this number this is for the case where we started fully developed flows in the fully developed both thermally and hydro dynamically fully developed region 3 and constant wall temperature boundary condition this was the Nusselt number okay so now that we are getting as a asymptotic solution to the thermal thermally developing case for large values of ? the same same value so this is here fully developed okay that comes out naturally as an asymptotic solution now wrap up this particular case as you can see this term Louisville problems any general term Louisville problems you cannot find direct closed form solution you have to do a numerical solution and therefore sometimes it is difficult to do so so once the numerical values are calculated and tabulated there was a person called housing who actually fitted the empirical curves to this numerical solution for different values of N he took and then he substituted them there and then he fitted set of numerical empirical curves to the numerical solution and then he directly proposed an empirical correlation which is independent of all these constants and the eigenvalues okay so that is a very famous correlation so that goes as a new is equal to 3.66 plus 0.0668 into grades number divided by 1 plus 0.04 grades number to the power two thirds okay where your grades number is defined as 1 over or we can say 1 by grades number is basically X by D by Peclet number okay so this is a very simpler correlation you just substitute the axial location non-dimensional form as a crates number into this expression and you directly get your local Nusselt number okay you do not have to find out eigenvalues the corresponding constant corresponding to n equal to 0 1 2 3 for this okay this is an empirical correlation which fits very well with the exact solution okay so you can see for the limiting case where your X by D goes to very large values your 1 by grades number is basically see here your grades number becomes very small for large values of X by D so then this particular term here disappears and it will lead to the limiting case of 3.66 which is the Nusselt number for fully developed flow okay so if you want to just plot the axial variation of the local Nusselt number with respect to 1 over grades number so you will find that this is for the constant heat flux case anybody remember what is the asymptotic solution 4.4.3 4.4.3 roughly this is for a constant heat flux and this dotted line here is for constant wall temperature and the asymptotic case leads to the value of 3.6 okay so if you plot the local variation you know once you have the first five dominant terms you plot it as a function of 1 over grades number and you will find that exponentially it is a decaying function okay and also you can plot both the constant wall flux boundary condition case constant wall temperature boundary condition case the constant wall flux case as a slightly higher Nusselt number and asymptotically that will reach to a value of 4.3 and constant wall temperature case reaches to 3.6 okay so this is to give an idea about the thermally developing region now as I said how do we get the constant heat flux case okay so that is a slightly different problem okay the problems that we have solved in the class corresponding to constant wall temperature boundary condition okay and parabolic velocity distribution this was an extension which was proposed by sellers extension of the great's problem now the other extension is for a non uniform wall temperature that is for case where you can have an axial variation of wall temperature which is a linear variation or you can also have a constant heat flux boundary condition so these were proposed also by sellers and I have uploaded document on the Moodle which gives the extended solution for the case of a plug flow the velocity profile is plug flow but the boundary condition is a constant wall flux boundary condition okay so there I will probably in the tomorrow's class give you hints how to approach that problem there the boundary condition is non homogenous because you have a defined heat flux so the thing is how do we homogenize that boundary condition for getting an eigenvalue problem so that that is the key once you know how to do that after that the rest of the things are straightforward so to get an idea you can just look into the Moodle where I have posted for the case of parallel plates okay that is a much simpler case to deal with the Cartesian coordinate system and taken case of plug flow and constant wall flux condition okay so I will give you I will just describe the procedure briefly and you can go over the document and you can do the same thing for the case of circular tube balls okay so with that the developing case will be over the thermally developing case or the thermal entry length cases will be done and finally we look at the case of simultaneously developing simultaneous entry length problems so those problems we cannot do analytical solutions because you cannot neglect any terms strictly speaking okay we have to go for a full numerical solution to the Navier-Stokes equations there have been some solutions where some approximations have been made but still they were involving a numerical solution so I will just give you the empirical correlations coming out of those solutions and I think tomorrow we should be able to complete this part and the last two classes on Thursday and on Saturday we look at the approximate solution to the internal flow problems so how do we use the integral method here okay.


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