Good morning, everyone. Day eight is here. We have been in this for eight days. This is incredible. How's everybody feeling today? Drop that in the chat below where you are watching from, how your fast is going. If you're still walking strong these eight days, let me know how you're feeling physically, what the Lord is doing right now in your heart and in your life. Some people are saying, hey, it's my day six. Pakistan is in the house. San Diego, India, still good. Great. Okay, Gina, praise God. Feeling okay. The body is weak, but the spirit is stronger. I am on day nine. Amen. Great water fast. I'm so much better. Legs were hurting yesterday, but I'm better today. I'm feeling good here today in South Africa. Good morning. Walking strong. Romania is in the house. God is blessing my soul. Amen. It's great. Anybody dealing with gout during the fast? Somebody's asking a question. End of day eight in Romania. So Romania, you guys are already done with day eight. We are just starting with a day eight. Fresno, California is in the house. Other people are joining in. Amen. This is awesome. Philippines is in the house. All right. So I'm assuming the stream is coming through. Everything is well on your guys' end and on our end. So today is day eight. Before we go anywhere, help me out right now and just, would you just hit thumbs up to this live stream on YouTube and on Facebook as well. Just click share. This will just help us to reach more people. It will help with the algorithm. If you have not done so, as well as if you are just tuning in for the first time and you are not doing the fast, my encouragement today is for you to join. We still join 13 days left and, you know, we're almost halfway through it and the time will fly like this. You won't regret it. And so go ahead and just honestly make up your mind today. A few things that I want to share with you from the beginning as I do every single day. If you have not gotten the book fast forward, go and go to Amazon and get it. I want to say huge thank you for everybody that's been leaving reviews on Amazon. I want to read one or two reviews just to kind of show appreciation and to encourage you guys to leave a review. So if you have not left a review, make sure that you go to Amazon and leave a review. Daniela Porta said, the subject of fasting has been in my mind in the past few years a lot. I fasted on and off in the past, but I feel like I'm starting to understand fasting in a fresh new way by reading this book. This book is practical, very concise, easy to read, easy to understand and implement. Thank you Vlad for taking time to write a powerful tool for life. Juliana, who actually purchased the book, she says, let me tell you this book is great, reading one chapter daily as you fast. It's been helping me. Pastor Vlad is anointed, man of God. Thank you again. And then Alessia and then so many others. So a huge thank you. If you left a review already, could you drop number one in the chat? But if you only got the book and you haven't left the review, drop number two in the chat. I just want to see our fans and our supporters over there in the live stream. And so I appreciate you each and every one of you. Wow. A lot of you already dropped the review. You guys are faithful. You guys are committed. A few of you have not, but I see that most of you have. Thank you so much. Much, much. Thanks. Amen. Amen. The other part that I wanted to highlight is that if you are not reading through the book online or physical book, you can actually have a reading plan, a fast forward reading plan with your friends and with your family and read it together. It's a great resource that you can give it a shot and stuff. So I just want to kind of encourage you to take advantage of that. If you're looking for something for your children, we also have it in the description below. And it will be a great blessing to you. If you're one of those people that you have a hard time reading right now, what you can do is you can actually get the audio book and part one introduction, introduction, fast forward.  So you can just go to Audible and subscribe to Audible, get the book for free, and then you can unsubscribe from Audible afterwards if you want. And so, yeah, that's also an opportunity. Appreciate you guys each and every one of you. Today is day eight of the 21-day fasting challenge. Personally, I am feeling a little bit weaker today than I felt yesterday. I felt like yesterday I felt a lot stronger than today, partially because I had a lot of meetings. We had three services yesterday, as well as I was live streaming after that. And then we had one more meeting with our, some of our leaders. And that lasted until like 10 p.m. So I was, woke up more tired today than before. But I think it's just physically, Sunday is a very heavy day for me. I call it my Super Bowl. It's a day where a lot of work, Monday and Sunday are the most challenging days. But I'm going to pace myself. And, you know, after spending some time in prayer, spending some time in the word, praying in tongues, you know, I feel refreshed in my spirit, but my body is a little bit more weaker today than I would say was yesterday. And so, which is okay. It's not about, you know, the physical strength always be there, but for the spiritual strength to be there. What I want to share with you today is a verse from Book of Daniel. And then afterwards, I'm going to share a little bit about the Daniels fast. But first of all, what I would like to go to is to go to Daniel chapter 10 and verse 12. So if you have your Bible, let's drop that in the chat. I'm going to go straight for this message that I have in my heart today to share with you. Daniel chapter 10 and verse 12.  Daniel, I would say Daniel had two types of, if I would say even a fast, it wouldn't be biblically accurate. But he had this first time where he did a Daniel's 10 day test, I would say Daniel's 10 day test, where he just got into the courts of Babylon. And then as a young man, he refused to participate in this diet that these people had. And so, God blessed him because of that. And then later on, we see that Daniel takes three weeks for mourning and pressing into God. In fact, if we just go in this chapter a little bit earlier, where the Bible says that, in those days, I, Daniel, was mourning three full weeks, so 21 days. I ate no pleasant food, no meat or wine came into my mouth, nor did I anoint myself at all, until three weeks were fulfilled. So this is where most of us get this idea of the Daniel's fast from this verse, 21 days. Now, there is no reference in here or no mention in here of the word fast. In fact, fasting means abstaining from food. This is more of limiting yourself from food. But nevertheless, I want us to look at the fact that though Daniel did not technically, biblically fasted, he did sacrifice. And he did go through self-denial. And God honored that. And so, for those of you who are doing that in these 21 days, I want you to notice that God sees your heart. Nowhere in the Bible do we see God telling us to do a Daniel's fast. In fact, for the record, in the Old Testament, God only commanded one day of fasting, but seven days of feasting.  God is not, you know, for starvation. Let's get this out of the way, right away, because some of us are kind of like think that God doesn't want us to eat. It's not about that. And so, and when you're considering doing, or if you are switching, or you are doing a Daniel's fast, I would just give you one simple rule. Whenever you begin a fast, remember, if it doesn't mean anything to you, it does not mean anything to God. Do not offer God something that costs you nothing. For Daniel, this meant something. Now, he was still working most likely in the palace and this wasn't just not eating. I want you to see he was mourning. There was brokenness that was taking place. He didn't eat pleasant food. Some translation actually says choice food, fancy food, rich food, food that satisfies, bread of desires. So, he gave up enjoying eating and only ate what was necessary. So, this was an expression of abstinence for the purpose of self-discipline. And this was no meat, no wine, and he didn't even anoint himself, meaning honestly, he went through three weeks of a lot of mourning, a lot of crying, a lot of repentance, a lot of pressing into God, a lot of humbling himself. Now, he didn't abstain completely from food, but he abstained from a lot of food, meaning he restricted his diet and God blessed him for that. God saw his heart for that. And so, if that is where some of you are at today, where you are not completely doing a water fast and you're restricting yourself from, you know, maybe eating just one time a day, or maybe you're also on the top of that, you cut off all of the other ungodly things in your life and you're really just pressing into the Lord right now. And while that might not be a biblical fast, it is a biblical way of denying yourself and humbling yourself. One of the very effective ways, and it could be a very practical way for many people who are not able to do a water fast for 21 days or 10 days or, you know, seven days, whatever that is for you.  You know, don't offer God something that means nothing to you. Like, if giving up coffee for you is not a big deal because honestly you don't drink coffee, that's not a sacrifice. It would be like me saying, I'm going to give up smoking. Well, I don't smoke. And plus, smoking is not a sacrifice. Smoking is a violation of your health. And so, sacrifices are things that actually mean something to you and they are good things. And we're giving up the good for the sake of the great. We're giving up the good for the sake of God. We're giving up the, I would say, the bowl for the sake of a blessing. We're giving up the physical for the sake of the spiritual. So Daniel does that. So if we keep on reading, we see that he gives up pleasant food, meats, you know, sweets, most likely dairy. And he gives up all these extra drinks.  And I told him, I said, you know, if you want to do that, why don't we join the fast? Next 14 days, do a fast, some kind of a fast. I'm not, I can't tell you what the fast, but I was like, for sure you should fast junk food and soda and sweets. And I was like, cut off these three things. Even after the 14 day fast, I was like, I promise you, you will experience breakthrough in your life. And I was like, in these 14 days, I want you to begin to pray down and break the stronghold of sugar, junk food, and obesity. It's a stronghold. And I said, God can help you to do that. Because I'm like, I can't just pray for you. Hey, let me break off the spirit of obesity. You know, let me break off the stronghold of obesity. That's just not how that works. If you don't have a habit to maintain that. And so I want you to see that Daniel, he doesn't do it for health reasons. He does it for spiritual reasons, but it does have health benefits. And so Daniel stays away from all of these things for 21 days. And then I want you to see this on the 24th day of the first month, I was on the side of the Great River. So Daniel spending some time in the nature, which I want to encourage you, begin to take some time. Don't just, even if you're praying and fasting, go a little bit on the nature, go for a walk, go in the park, get some fresh air. Daniel was on the side of the Great River. He lifted his eyes and looked and behold, there was a man clothed in linen whose waist was girded with gold. And then then it describes this incredible being. And verse seven, I Daniel alone saw the vision, but the men who were with me did not see the vision, but a great terror fell upon them. So he has this crazy vision. And then he hears the sound. And I want us to look at this, is that of course, this man touches him, he gets trembling in his knees. And he says this in verse 11, Oh Daniel, men, greatly beloved, understand the words that I speak to you, stand upright, for I have now been sent to you. Oh Daniel, a man, greatly beloved, I just want to speak to over somebody here today who is humbling themselves, who is cutting away food completely or restricting some foods and who is breaking themselves before God.  You are beloved of God. I want you to hear this right now. God is for you. He loves you. Drop that in the chat right now. I am his beloved. I am loved by him. Somebody dropped it in the chat. I am loved by God. Fasting doesn't make God love me more. Not fasting doesn't make God love me less. But I love this that after 21 days, this supernatural being comes to Daniel and says, Daniel, you are loved greatly. I love you. God loves you. He loves you. And that's an important reminder. You might not feel loved right now. You might feel discouraged right now. You might feel weak and tired. You might feel even a little bit confused. In fact, I'm pretty sure Daniel physically didn't feel super great. But when these words came, it brought such an encouragement. You are greatly beloved, greatly beloved. You are greatly loved. You are loved by your father. He loves that you are pressing in. He loves that you are giving things up. He loves that you value him more than even your own essential needs. He loves you. And then verse 12, he says, do not fear. The second thing I want you to drop in the chat. Do not fear. Whatever is about to happen in your life, whatever this year is about to bring, whatever you're going through, I want you to hear this from the God's Word and me speaking prophetically into your life right now. You are greatly loved, the first thing. And secondly, do not fear. There's over 364 or five times in the Bible it says do not be afraid. Why do not be afraid? Because God loves you and perfect love drives out fear. You have absolutely greater reasons not to be afraid. You may have reasons to be afraid, but your reasons not to be afraid are greater. Do not be afraid of tomorrow.  What's gonna happen with my family? What's going to happen with my marriage? What's going to happen with my business? What's going to happen with my ministry? Am I gonna be able to carry this load that I've been entrusted? Am I going to be able to finish the race strong? I'm not sure what to do next. I'm confused and anxiety can settle in. If you keep meditating on the things you don't know, if you keep rehearsing the things you can't control and God comes to you today through his word and he says do not fear. He said that to Israel when they were walking through the to the Red Sea he says don't be afraid. The enemies you see today you will see them again no more and God split the Red Sea. The angels came to the shepherds and they said don't be afraid. I bring you a good news. Angel came to Mary and said don't be afraid. You will conceive a child. An angel came to Zechariah and he says don't be afraid. Your prayers have been heard. Do not fear. Jesus said when we get persecuted do not fear those that kill the body. But there's nothing else they can do. Don't fear tomorrow. Don't fear the future. Don't fear the ministry. Don't fear fasting. Don't fear sacrificing. Don't fear giving your life fully to God. Don't fear that oh who's going to provide for me? Who's going to care for me?  Somebody just one more time dropped it in the chat. Do not be afraid. Only believe. Only trust. If you can't see it trust them. Blind Bartimaeus couldn't see Jesus but he could hear Jesus and sometimes when you don't see the future hear God saying to you don't be afraid. I got you. I'm holding you. I'm walking with you. You're in the palm of my hand. I will see you through. You're going to make it. Now then the angel said this spiritual being said from the first day that you set your heart to understand and to humble yourself. Your words were heard. The third thing I want to share with you today is this.  Some of you just need to be reassured of this. Number three dropped it in the chat. God has heard my prayer.  Not only God heard your prayer. I want you to see what the angel says. He says the first day the first day you made a decision to fast. The first day you made a decision to pray. The first day you made a decision to humble yourself before God. God saw that. God there and then saw that. And he heard your words. He heard the cry. He heard the attitude. He heard the posture of your heart saying Lord I want to know you. Lord I want to seek you. Lord I want more of you. Lord save my family. Lord I need a breakthrough in this very difficult time. And the angel said I heard you and I have seen the first day you made the decision. The first day you set your heart. That's why fasting is so important because it's about setting your heart. It's not about just abstaining from food because once you set your heart right God sees that. God is moved by that. God takes notice of that. And God hears that. Your humility. You're setting your heart right and you're pressing into the Lord. So the first thing I mentioned you are greatly beloved. The second thing do not be afraid. And third thing when you set your heart to seek Him He sees you and He hears you. He doesn't hear you when you get the answer. He hears you when you pray the prayer. Somebody dropped it in the chatter. Now that's the word. God hears you not only when you get the answer. He hears you when you pray the prayer. He said to Daniel the moment you made up your mind to humble yourself I heard you. He got the answer 21 days later but he got the prayer. Prayer was received 21 days earlier. God hears your prayer when you pray. God sees your humility when you fast and He dispatches His angels your way. Drop that in the chat. God heard me before I even got the answer. God saw me before I got the breakthrough. God hears you when you pray. Not only when you receive the needed answer. Sometimes we feel like I prayed but I don't see the answer. Please understand God hears your prayer even if you don't see the answer. Even if you don't see the answer to that prayer for 21 days all Daniel was doing was crying morning. There seemed to be no answer but the Lord came and He says I want you to staple this verse in your heart. From the first day you've set your heart to understand your words were heard. Your words were heard. God hears the prayers when even we don't see the answer. Those prayers are heard. Now rises the question. Why am I not seeing the answer? Before we answer that I'm gonna touch on one more thing. He says I have come because of your words. The fourth thing I want to encourage you with today is that God is working behind the scenes. God is coming with an answer. He is working things behind the scenes you don't see. He is right now working in the dark. Maybe you don't see. God is arranging things.  He says I have come because of your words. God dispatches his angels because he heard your prayer. God releases his favor because he saw your humility. God is releasing his breakthrough because he saw the brokenness in your heart. He's moved by our faith. He's moved by our prayer. The need doesn't move God. Your faith does. Your prayer does. Your humility does. Your brokenness does. It's not the fact that you're not putting food into your mouth. It's that your heart is set because fasting sets your heart. What did the angel say? He says you set your heart to understand. You set your heart to humble yourself before your God. So he's on the way to you. So the first thing I mentioned, you are greatly beloved. The second thing I mentioned is do not be afraid of the future. The third thing I mentioned is God heard your prayer even before you see the answer. God heard your prayer even before you have witnessed the answer. The fourth thing I mentioned is that God has dispatched his angels when he saw your heart posture of humility. He is working behind the scenes. Now the last thing I want to mention and that's number five. That's verse 13. But the prince of the kingdom of Persia withstood me 21 days. I believe one of the reasons why some answers are delayed is because there is a warfare going on in the spiritual realm that you and I are not aware of. Sometimes God pulls the curtains and lets us see the warfare, but usually he doesn't. And Daniel persevered for 21 days, not knowing there is a warfare taking place. The answer was on his way. Prayer was heard. Angel was dispatched. But he said, but the prince of the kingdom of Persia withstood me 21 days. And he said, and behold the Michael, one of the chief princes came to help me as you persevere, as you persist. I believe that there becomes an overwhelming victory in the spirit realm that you will benefit from in the natural realm down the road. It might not be right away, but as Daniel persevered these 21 days, help came in the realm of the spirit. Help came to this prince that was fighting on his behalf.  I want to encourage you today. Don't get discouraged when it gets hard. Don't quit when it gets difficult because there is a warfare that's taking place. Some answers, when God hears the prayer, he sends the answer right away and they are received right away. But sometimes there are delays that happen and we should not be discouraged by these delays. We should anchor ourselves deeper in God and press in further, go deeper until we get that answer in our spirit that something changed, something shifted, something is going to be different from now on. If you quit too early, if you give up too soon, if you kind of not persevere, the danger of that is that you pretty much can sometimes, not in every case and I don't want to bring guilt trip on anybody, but some of us, we abort the miracle. We abort the breakthrough because we quit when it gets hard. Some of us, we quit when it gets hard in our faith in God. The Bible says that the seed that fell on the ground and the hardships came in and they just choked the seed. They just didn't, the seed just dried up. People just give up too quick.  God loves you. Don't be afraid. God is working behind the scenes even if you don't see it. God heard your prayer, but if there is a delay, if there is a resistance, if there is a withstanding, the Bible says that this being withstood against the angel that was coming to help Daniel, don't quit. Paralyze the resistance with persistence, persevere, drop that in the chat. Paralyze resistance with persistence, persevere, don't quit, persevere, keep going and then the understanding will come, breakthrough will come. God is working behind the scenes on your behalf. God is working behind the scenes on your family's behalf. God is arranging things. He's working things behind. Your task is to stay humble. Your task is to keep praying and your task is to be obedient in what He tells you to do, whether it's in the fast, whether it's in giving, whether it's in forgiving, whether it's in intercession, whether it's in whatever that is, your task is to be obedient in that. Amen. I believe that this was a blessing to a lot of us. If this was a blessing to you, drop number one in the chat right now. If you receive this word, these five things that I've shared with you today, drop number one in the chat. If you are just tuning in, there's hundreds of you that just recently joined in as I started to speak. Don't forget to hit thumbs up right now to where you're watching and then share this broadcast still. I know that we're coming toward the end, but share this broadcast still. If you're just tuning in, I'm going to take a little plug in right now and let you know that I have a book called Fast Forward. It will really motivate you in your journey of pursuing God. Make sure you check that out on Amazon. You can download it on my website as well. And if you have gotten the book, be kind to leave a review. It will mean a lot to us. You can get this book in digital form, in audio form, as well as in the paperback cover. And so that's just my little plug in. Thank you so much for that. And right now I want us to take time to pray. Lord, just drop that prayer emoji where you are watching right now. Drop that prayer emoji where you're tuning in from and let's pray. Let me pray for you. And then we're going to answer some questions as well as read the health tip for today. Okay, prayer emojis are coming through. Come on. Amen. Let's pray. Dear Heavenly Father, we believe your word. We believe that you are working behind the scenes. We believe right now that we should not be afraid. We reject fear. We will starve fear. And we will feed our faith. We will look to you, Lord. Lord, right now we make a decision to accept your love. We're not more loved because we're not eating. We are loved because Jesus died on the cross for us, because your love. Lord, right now we renounce fear, anxiety, discouragement. It's just all of these demonic confusion and doubts. Lord, we choose to believe that you see us. When we set our heart right, you notice us. When we humble ourselves, you hear us. We believe that you have heard every prayer we prayed. And we believe that you are answering these prayers. Lord, we believe that you are sending an answer our way. We believe, Lord, that you are also sending angels to bring breakthrough in our life. You're working on our family members. You're working on our marriages. You are working on our finances and on our ministries. Lord, we trust in you. Even when we don't see it, I know that you're working. We trust that you are working. We love you, Lord. Strengthen the people that are fasting and praying. Would you purify their hearts? I pray for those who are unsure whether they should do it. Would you just put it into their heart to join this fast, even though it's, we're almost halfway through. Given the desire, draw them. Let them not offer something to you that doesn't cost them anything. As you met Daniel, meet us. As you answered Daniel, answer us. As you have come through for Daniel, come through for us, Lord. May we experience breakthrough that we so desperately need in Jesus' name, in Jesus' name. Amen. Amen. If you receive that prayer, drop in the chat I receive, and right now I'm going to go to the fast forward book. So in the fast forward book, day eight, which is today is fasting is feasting. Come on. Fasting is feasting. So read the devotional for the day eight. It deals with feasting. So make sure that you read that. It will be a great blessing to you on how to feast during fasting. You're like, so how can I feast during fasting? Read the devotional and then the scriptures to meditate. So I have the book on this right there, open where you can find it or on my website. But let me read to you right now the fasting tip for day eight. During the stage one and two of the fast, your body will still be expelling toxins and damaged cells every time you go to the bathroom. Using an intestinal cleansing product will help more thoroughly cleanse and detoxify your body. So remember, as you're going through right now, this time, your body is still cleansing itself and expelling toxins and damaged cells every time you go to the bathroom. So that's one of the reasons why sometimes we experience different color in the urine. So stay hydrated, drink water, add a little bit of sea salt or fasting salt into your water so it could help a little bit with you. And yeah, so we will take just a few moments and answer some questions and then get ready for our live stream tomorrow. Amen. So I've got about 100 people watching in Hungry Jen as well as on Facebook. And then amen. Guys, as I'm going to just open the questions, I'm going to ask you for something. If this was a blessing today, if God really spoke, I want to encourage you to also give toward the ministry. I want to encourage you to give also toward last subject ministries for a few reasons. We as a ministry is a good soil where we release books. Like if you check right now, I'm going to show you on our website, we release content. So tons of books that are released and all of them are available free of charge, from fast forward to spirit fill Jesus, fight back, single ready to mingle, break free. Like look at how many translations we've done there for the glory of God. And so as well as we have so many other things that we do like courses. So not only we do books, but we also release courses. And as you know, we offer them free of charge as well. People deliverance, Holy Spirit, so much, so much things that we produce here at this ministry, release blogs, as well as other resources. So not just blogs, but podcasts weekly. Actually, right now we're doing daily podcasts, as well as PDFs where you can download longer teachings, tons of reading plans for people, as well as audio and so much. So just wanted to say guys that this ministry is a good soil. We love the Lord. We take these funds very seriously to put them into distributing the message of Jesus and to spread the gospel of the Lord. And so you can give through cash app, you can give through Venmo, you can give through PayPal, but most importantly, you can give through the website.  And then from there, you can actually monitor your giving. You can actually change your giving, cancel your giving. And so and then partnership is where we are looking for more partners this year. And maybe that's one of the things the Lord will put on your heart this year, not just to occasionally give one time, but to become a partner. We have about, I would say think about 10 people that are on the on the staff for this ministry. And so your giving really helps also their families.  Some of them are from Tri-Cities. Some of them are not from Tri-Cities. And so I just wanted to let you know that this is what it goes for. And I really appreciate you guys making that. If that's what the Lord puts on your heart to become a partner, if God puts on your heart, may this be one of your decisions this year for next 12 months to partner with our ministry monthly. You know, and you might have a lot of things that you partner with like Netflix and other stuff.  Amen. So thank you so much. So many of you started to give on YouTube chat.  God bless you. Kosoa. God bless you as well. Donna. Amen. Amen. God bless you. God bless you guys for that. And those that are giving online as well, or cash up or Venmo, we really appreciate that. We honor you and we celebrate all of our donors and all of our partners. Now, when it comes to questions, how many times is it written in the Bible? Do not fear. I think it's how many days how many days is in the year 365? So 365 from my understanding is 365. So what in intestinal cleaning products do you recommend? So I don't have any recommendation person, I would just say that you just search that for yourself. Things that I don't necessarily personally use anything of that, but it could be helpful and to do that. So just I would just ask that you just do some research yourself. When coming out of a water fast, can you drink protein shakes for the first three days? I don't know much about protein shakes. So I would just research that more. I stay away. I just drink soups and eat just bone broth and some more soups than protein shakes. That's just personal kind of approach. And I know a lot of people who do that. So I would just do a little bit more of research. I've never drink protein shakes after the fast to come out of the fast. I'm doing a water fast for first three days. I'm awake for 24 hours, then sleep to three hours up again for 24 hours. Is it normal? Do I need to do anything different? Actually, for some people, it's normal to have your sleep gets a little bit disturbed during the fast. So I would encourage just use compound. Let me see if it's a compound because last time I said vitamin B and it's the wrong one. Super B complex with electrolytes. I would just begin to add that into your take one tablet a day. Super B complex and then fasting salts and that could kind of maybe help with that. But if you're not sleeping at night, just spend some time in prayer.  But I've had, when I was doing a 40-day fast, I've had this for about a week, when I struggled to sleep at night. I couldn't sleep until like two o'clock in the morning and I'm an early sleeper. So like I go to sleep very early and I just kind of read and waited till my body got tired and I would sleep. I would sleep less. It seems like my body would wake up way earlier and I felt fine. So I just continued. I didn't do anything. I just kind of weathered that. Should I be expecting something from the Lord even if this is not a request fast? I dedicated this fast to get closer to God and to humble myself. I didn't ask for anything. Yeah, I think, you know, tomorrow I feel like I want to share that about the importance of when you don't ask God for anything, how God will answer still. This is not to discourage us from asking God, but at the same time, I'm going to share about this tomorrow because you have not asked. So I'm going to share about that tomorrow. So just don't miss tomorrow. Getting perverse dreams while fasting. This could be a way of God showing me what to pray for. Yeah, or this could be a way of God showing you what to begin to target specifically in your prayer. Could you share how your 25 to 40 day went on your 40 day? So from 25 days to 40 days, I just came from Ukraine and so I had an international speaking trip and it was, I was a little bit more weaker on some days. So I paced myself because I don't have a very physically demanding job. I had the freedom to rest more when I needed to and I continued too fast. I'm going to tell you that I did get discouraged sometime from 25 to 40 day to quit. It was a little bit discouraging. I had these thoughts like just it felt like it will never end and all of that. But I didn't let them take root in my heart. I just persevered physically. I experienced sometimes knee pain when I would sleep and then sometimes I had a hard time falling asleep for a few days. But I just persevered and then I bought the fasting salt so I added that little powder into my drinks every few days and that's about it. But overall it was just normal. It was just normal. It did get a little bit discouraging around 30th day to kind of like oh I just want to quit and so but I just I knew it was in my head and so I didn't give it too much attention and thought and I tried to distract myself when those thoughts will come by reading by just doing something else. So yeah. Are we supposed to be sexually pure during a fast? I've just recently married so I've never had to deal with this in the fast. Now good question. In the Bible it does not tell us to abstain from sex in marriage when you're fasting. It says to abstain for a season of prayer and again it's not required. It's not required. It's just it's just an option. So personally there is no biblical mandate about abstaining from sex when you are fasting. Fasting is abstaining from food. Okay not from sex. So when you are married physical intimacy is part of a good healthy relationship and it should be maintained and it should be and it should not be neglected and so this has nothing to do with that. So I see that as no bearing. Now if the Lord puts on both of your hearts and you both agree that you want to abstain from that that's completely fine but at the same time fasting is not abstaining from sex it's abstaining from food. So that's just what the scripture teaches us.  Being sexually pure is having your thoughts pure, having your motives pure, having what you look at pure and not having intimacy with other people physical intimacy. So that is purity. There's nothing impure about a husband and wife being together as husband and wife and so if your thoughts are impure, if you're watching pornography yet then you're not pure but godly physical intimacy is pure. All right guys I am going to wrap this up. It's been an hour already. Tomorrow we are going to go just a little bit deeper and a little bit further and deal with the component of people which I believe will be very helpful for people who are fasting specifically for honestly drawing closer to God. They don't necessarily have a particular big request and so I'm going to give you a lot of encouragement tomorrow for that so make sure that you join tomorrow. You don't miss tomorrow. We appreciate you. I see somebody is asking about the book so it's a fast forward book that goes along with our fast 21-day devotional. Each day it has scriptures, prayers, health tips and there's an audio version for it. There's a paperback, there's a Kindle and there's a hardcover. Now right the way I'm going to tell you the hardcovers are pricey because it just takes a lot for Amazon to make them and so they charge a lot and for people like me they don't give me an option to actually lower the price for it. It has to be like the minimum of what it costs them so if you already got this book make sure that you leave a review. Much appreciated. If you haven't got it just go on Amazon and get it today. Now if you haven't gotten my other books like Break Free, Fight Back, Single Ready to Mingle, Spirit Fill Jesus make sure you check them out. They will be a great blessing to your walk with God. Amen. Appreciate you guys. God bless you. We'll see you again tomorrow.


Well, extra enforcement during a holiday DWI campaign resulted in more than 2,600 driving while impaired arrests. More than 300 law enforcement agencies across the state took part in the campaign, which ran from November 22 through December 31. The drunk driving arrests include the following. The Fairmont Police Department reports a driver arrested for DWI on Christmas Eve was arrested again on New Year's Eve for DWI. Both times, she was driving on the wrong side of the road. The Onamia Police Department reports a woman who was going more than 100 miles per hour and side-swiped another vehicle was arrested for DWI after a preliminary breath test of .36, four times the legal limit. And the St. Cloud Police Department says a driver who refused a breath test recorded a .24 blood alcohol content the following day as he was about to appear before a judge. During the campaign, 12 agencies reported a blood alcohol level of .35 or higher. If you enjoyed this segment of Lakeland News, please consider making a tax deductible contribution to Lakeland PBS.


Good morning. So, we will quickly revise what we have learned in last lecture. We are looking at models for non-ideal reactors. You know non-ideality comes because of the non-ideal flow patterns and we have to somehow characterize this non-ideality. So, we are looking at one parameter models. Before this, in the earlier chapter, we looked at zero parameter models with two extreme situations, maximum mixingness and complete segregation. Now, in this particular chapter, we are looking at models with some parameters involved in it and these parameters have something to do with the flow patterns inside a reactor, the extent of mixing and so on. So, we will just moving away from zero parameter models. The problem with zero parameter models I told you that gives you bounds on conversion, does not tell you the exact conversion. Whereas, now we are looking at a model with a parameter. So, it is going to give us some idea about a conversion, exact conversion. The model that we are using is right. So, we are going for a simple model. Now, it is a one parameter model. You can have a model with many parameters, but right now you are just looking at one parameter model. The two types of models that we have looked at. The first is tank in series and the second one is dispersion. Of course, we have just looked at this one in detail and today we are going to talk about dispersion model. But just to quickly revise what is tank in series model, we have a tubular reactor, say tubular reactor and there is a flow that is taking place in this direction. Now, a plug flow reactor is a reactor where the mixing in radial direction is maximum, but there is no mixing in axial direction, right? No mixing in axial direction, but then velocity profile is flat and if I have a pulse injected here or a tracer injected here, then I will see the same tracer after a time tau here. That is the e curve and something that is known to you now. Now, if this pulse gets disturbed when it comes out, then it may be because of some extent of back mixing. So, if it comes out like this, there is a possibility that some back mixing is occurring here. It is not just a plug flow behavior, but it is hampered by some kind of back mixing. This may occur because of the concentration waves moving or there might be some disturbance in the flow, internals for example, say part column whatever, but I want to characterize this extent of back mixing. So, what I am going to look at is this as a series of small small CSTRs and this n number of CSTRs and this n characterizes extent of axial mixing. So, this n is a parameter, a given tubular reactor if I say that I have n number of CSTRs in series which are equivalent to this tubular reactor, then n becomes a parameter. This n can be 1, 2, 3, 4, 5, 6 and infinity. So, infinity means it is a plug flow reactor and n is 1 means it is a CSTR because complete mixing and in between these two you have partial back mixing and this n will n is large means this less mixing and this small means this good amount of back mixing that is occurring. Then we derived equation for n in terms of, now how do we get a value of n? n is obviously obtained by the tracer experiments. So, tracer experiments is a tool for any given reactor under the conditions whatever flow rates and all we are operating it at that tool gives you the parameter that is nothing but n. So, if I get E curve the variance in the E curve get tracer experiment the E curve and the variance in the E curve you know what is variance. Sigma is equal to 0 to infinity T sigma square sorry T minus tau square ET dt. So, variance now this variance is related to n what is the relationship n is equal to we already looked at it tau divided by sigma square. Now what is tau? Tau is a residence time total residence time volume of the tube divided by the volumetric flow rate volume of the tube do not talk about do not confuse it with the volume of small small CSTRs. So, given a tube somebody ask me whether this will behave like a plug flow reactor or not tell me this is flow rate that I am operating it at what will I do I will do a tracer experiment. I will see whether I get a nice pulse just getting repeated or all very close to something that I had injected after time tau if it is not happening it is not a plug flow reactor. If it is getting dispersed as I showed before it is a slightly dispersed or the pulse get distributed rather over time in that case there is definitely some back mixing occurring and there is a way to characterize this back mixing. I will do a tracer experiment I will get E curve from E curve I will get variance variance because in this just E curve is to be known otherwise everything else is known. So, I get E variance once the variance variance is known tau is nothing but volume divided by volumetric flow rate I get a value of n is number of CSTRs. Now once you have a value of n you have x that is conversion for series of CSTRs given by tau i k square sorry not square is raised to n. Now tau i is nothing but tau divided by n right and k is rate constant this is for the first order reaction do not forget this. And for second order again I can derive some equation if not I will just go on doing calculations step after step. So, it has allowed me to get a conversion for the given value of n sorry sorry should be. So, this is a conversion for a tubular reactor that corresponds to n CSTRs in series. Now one thing I forgot to tell you last time is this n. Now after doing this exercise and getting the value of n n need not be integer n can be 3.5 it can be 4.75. So, this value of n now as such like really number of CSTR number means what should be integer. But just ignore or others forget it whatever value of n comes just put it here. And once you do that whatever value of x comes that is a real value of x which is close to what you would get in that real reactor which is a tubular reactor neither PFR nor CSTR. It must be noted that this is possible that is using the non integer value of n only for the first order reaction. For other orders analytical expression is not available for the conversion after n reactors in such cases n will have to be rounded off to the next highest integer fine. We will solve some example or I will tell you the methodology to solve example as we go ahead. But before that let us look at dispersion model. So, this is where I stop as far as the tanking series model is concerned. You know n is a parameter n is to be obtained once you get n you get a value of conversion for a given reaction for a tubular reactor. Now, let us go ahead and discuss the other model that is dispersion model. Now, once we have one model why to have another model this is our convenience sometimes some models work well sometimes they do not work. So, I probably after discussing dispersion model we will have we will spend some time relating these two models as well. So, there are two models available just we should know both of them. So, that because they are both of them are popular in industry. Now, dispersion model in this case dispersion is what dispersion is again a synonym for axial mixing that is occurring of course, it can be in radial direction also. So, just dispersion it can be in any direction. But as far as tubular reactor is concerned we are talking about axial mixing. But it can be radial also can club these two together and get overall dispersion all right fine. Now, again you have a tubular reactor if you if you inject a tracer because of back mixing is going to get something like this if it spreads it becomes flat in that case extent of back mixing is is is very large right this is a behavior that I am going to see. Now, can we give a mathematical treatment to this the first model that we looked at was assumption that you have series of CSTR tanks in series. But here I am not going to do that I am just going to look at the way I do it for a plug flow reactor take a differential balance take a differential balance. Now, for unsteady state suppose I injected tracer how it behaves with respect to time I have to write unsteady state balance a PFR is a DFF by dv is equal to R is a steady state balance right. Now, I have a tracer injected R term that is reaction term is 0 there is no reaction taking place a non-reactive conditions ok. So, you have flow taking place but at the same time tracer concentration at any given point is going to change with respect to time ok. So, accordingly I will have an equation ok. So, what is the equation unsteady state equation for a PFR. So, it is going to be DFT F T is a flow rate of the tracer normally we say F A you know for A that is a reactor now instead of A now I have T here. So, DFT by DZ is equal to AC into DCT by DT. So, this is a unsteady state balance for a tracer in the tubular reactor right tubular reactor why I am writing this balance I am giving a mathematical treatment to this as time proceeds how the things are going to change with respect to Z and time as well. So, this equation describes everything now what is F T what is F T in a normal situation F T now this is very important for a plug flow reactor what is F T we say that is only convection that is V into C that is volumetric flow rate into concentration that is for a normal reactor. But in this case it is the flow is not taking place only because of convection, but there is an additional term which is caused by axial mixing which is caused by dispersion. So, this F T this let us try and write the expression for this F T this F T is of course, U into AC into CT what is this this is convection right, but it is not just this apart from this there is an additional term that is coming out of dispersion. So, let me call this as F T D and what is F T D this F T D is because of dispersion now normally it is written as some coefficient we will talk about this later D A into AC look at look at how it look at expression it is quite similar to Fick's law this is quite similar to Fick's law D A AC into del CT by del Z right. So, without AC it is flux. So, I am multiplying it by AC what is AC AC is a cross sectional area right. So, this is the expression for the dispersion or flux due to dispersion and why it happens it happens because of concentration gradient is a front that is moving of a tracer is a front that is moving. So, it is a concentration difference that causes the dispersion this is analogous to diffusion and this expression is analogous to Fick's law. I am not calling this as a Fick's law because the coefficient that I am using here that is D A is not a regular diffusivity or is not a normal diffusivity is not bulk diffusivity that we define in the case of Fick's law this is something that combines all the effects. Now, you may ask me whether it has the capability to combine all the effects it may not have in that case your one parameter model does not work that is it. But if for some reasons of somehow if for a given geometry and flow pattern if it is able to take care of all these dispersion effects in one parameter then you are done. That means your model is perfect go ahead and use it and try and predict the conversion of the reactor for a given volume. So, that is the meaning of it. So, I am I am expressing the dispersion by this particular term. Now, you you may ask me what kind of flow pattern laminar or turbulent. Now, it it can be proved that this expression works well even for laminar flows this works well for turbulent as well. So, this can be a diffusivity in the case of turbulent flow or it can be in the in the case of laminar it can be molecular jumping from one layer to another layer which are otherwise not mixing. So, one layer moves at one velocity the other layer moves at another velocity in laminar flow and there is exchange of some molecules this is non-ideality. So, it is not exactly laminar, but then that case this particular exchange can be incorporated here in this term diffusivity. If time permits we will try and derive the expression separately for laminar flow and show that it can be expressed in this way. But right now let us go ahead let us not worry much about whether it is laminar or turbulent flow we say that this is applicable to any of flow that is laminar or turbulent.  Now, we have we already have this equation we have this equation and in this equation f t is given by this. So, let us substitute for f t here substitute for f t here remember f t otherwise was this only for a plug flow reactor now that we are considering dispersion we have an initial term. So, we have to put this here. So, if you substitute for f t what you get is this d a del c t by del z square. Now, you get a second order partial differential equation del u c t divided by del z is equal to del c t divided by del t. Second order is obvious because f t comes in we have d f t by d z and f t is now d c t by d z you have d c t by d z term in f t. So, that is why you have this expression. So, this is the expression that I am going to deal with and with this equation. Now, I have the tracer concentration changing with respect to time and z if I solve this equation I will get the response in sense rather this tracer how it spreads and all that at any given time and length in the reactor fine. So, let us remember this equation and in order to solve this particular equation I need boundary conditions. Now, these boundary conditions they are quite peculiar and we are going to spend some time in knowing what these boundary conditions are. Now, you have a tubular reactor you have the inlet and you have the outlet. Now, we are talking about b c that is boundary conditions differential equation needs boundary conditions otherwise you cannot solve this equation. Now, how do I look at it? You will say it is not that difficult what is the big deal is inlet concentration is outlet concentration and all that. So, if you see this is your second order partial differential equation how many boundary conditions you need here you need one initial condition and two boundary conditions right. So, there are some assumptions to be made here and some of the assumptions they work so well. So, you have a flow taking place you have a flow taking place dispersion is happening. Now, where is the dispersion happening? Dispersion is happening here that means d a value of d a is greater than 0 in the reactor. So, this is a this is a reactor this is a reactor. So, value of d a in this domain is greater than 0. Now, this is before the inlet and this is after the outlet. So, before the inlet I have a choice whether dispersion is happening here as well or not. So, let me let me let me consider a situation where d a is equal to 0 here and d a is equal to 0 here. The node dispersion is happening in the outgoing channel and incoming channel it all depends what kind of system you have what kind of nozzle you have here and here. But if I assume this then this is called as closed vessel and we have a corresponding boundary condition we will come to that. But this is a situation where I have a closed closed vessel that means there is no dispersion here and there is no dispersion here. There is another extreme this is another extreme where I have open open case or open open vessel where d a is greater than 0 here as well d a is greater than 0 here as well d a is greater than 0 here as well why not is sometimes like we showed the arrow like this and not straight why this is one indication where the flow is not ideal it is not like a plug flow rate. So, it all it all depends how we want to show it, but then the indication is like arrow is distorted means there is some back mixing happening all right fine. So, this is called as closed closed vessel. Now, you may have situation where you have open closed or closed open depending on what is happening. So, you have a choice and depending on that decide what kind of boundary condition is applicable in my particular case because every time you have a different expression. So, let us look at sorry did I say oh sorry this is not closed closed let me let me correct myself sorry this is not closed closed this is open open earlier was closed closed this is open open. So, you may have several possibilities and you decide what you want it can be closed closed it can be open open it can be closed open or it can be open closed all possibilities. So, depending on that you will get expression for the boundary condition. So, let us write expression for a closed closed vessel. So, let me draw it again at inlet you have flow taking place. So, whatever coming in is going out at this particular boundary right. So, I have expression f t tracer 0 minus here it is 0 minus and sorry minus and is 0 plus f t 0 minus at any given time is equal to f t 0 plus at any given time at that particular time of course, these two times are same all right. So, that is obvious no that is obvious there is no accumulation at the inlet. So, whatever comes in goes out. So, molar flow rate f t of tracer coming in is equal to coming in at a inlet is equal to coming inside a reactor fine. So, let us write expression for f t now f t in there is no dispersion here closed closed. So, there is no dispersion here. So, it is velocity into cross sectional area into concentration 0 minus t right concentration at inlet that is 0 minus at particular time I am multiplying it by u into a c what should I write here now what should I write here for this for this. Can I just write this u that is velocity a c c t 0 plus t is that sufficient the answer is no this is not sufficient because inside it is not just the convection or the velocity driven flow, but it is dispersion as well. So, we will have an additional term which is given by a c into d a right d c t by d z at z is equal to 0 plus right. So, f t coming in from here is this and f t here is this plus this right. So, these 2 concentrations are different these 2 concentrations are different concentration here is different concentration here why because this is an additional term which will make sure that is concentration changes. If this term was not there if this term was not there then u a c will get cancel in these 2 concentrations would be equal, but this term is responsible for making these 2 concentrations different alright ok. So, let us let us simplify this further c t 0 is equal to minus d a divided by u d c t by d z at z is equal to 0 plus plus c t 0 plus t. So, this is one boundary condition that is at inlet. So, inlet boundary condition I will say inlet pc at the exit or outlet what happens at the outlet. So, let me get back to this figure or let me draw it here again. So, d a is greater than 0 d a is equal to 0 here there is a continuous flow there is a continuous flow. So, at z is equal to L at z is equal to L c t at L minus is equal to c t at L plus the concentration here is same as concentration here concentration these 2 points are going to be same. So, what it means is d c t by d z is equal to 0. So, this is a boundary condition at the outlet. So, these are the 2 boundary conditions that I have alright. So, these 2 boundary conditions and where is our equation is the equation here. So, this is the equation this equation and these 2 boundary conditions what else do I need I need initial condition what is that initial condition at time is equal to 0 what is the value of concentration at all lengths. So, at time is equal to 0 initial condition I c initial condition at time is equal to 0 and z is greater than 0 that means at any length c t 0 plus 0 that means time 0 is equal to 0. Of course, it can be at any any length 0 plus means on word 0 initially the concentration 0 because I am injecting a tracer at time 0. So, at time 0 inside reactor concentration is 0. So, this is initial condition. So, initial condition then you have boundary conditions. So, these are the boundary conditions and this is your equation this is your equation main differential equation right. I can solve these equations together to get the tracer concentration how it changes inside a reactor. Why are we doing all this because we want to see how the tracer get tracer spreads. Why do you want to know how the tracer spreads because that is going to tell me the flow pattern inside a reactor. So, I will be able to characterize the flow pattern like what I did in the case of tanks in series. I characterize the flow pattern in terms of a number called number of tanks and similarly I am going to get a parameter here. We will see what parameter right that will characterize the flow pattern. So, let us solve this equation we will see right now see the main difference in what we did before and before means in the tanks in series model. There we just discretized it the entire domain the volume in different tanks. Now, I am looking at differential equations. So, there is a discrete dynamics there here it is a continuous dynamics. I am writing a differential equation here there I was writing difference equations. See the difference you have come across this particular way of formulating a problem in many cases the distillation column absorption column see number of stages packed column tray column similar. So, now let us go ahead before that we need to non-dimensionalize this equation because when we non-dimensionalize we will come up with the parameter that is going to decide everything. Now, what is this non-dimensional form of the equation? So, non-dimensional form of the equation is going to be like this. So, let us start with this sorry. So, let me this is a differential equation I want to non-dimensionalize this. So, let me define non-dimensional concentration psi C t by C t 0 lambda z by L theta. In fact, theta we will come back to this it will come on zone you know typically what happens when we do non-dimensionalization. If we do this. So, let us see you have d A d 2 psi by d lambda square. Now, here what we will see is you have L square minus u you can take it out assuming that velocity is not changing with respect to z and t. So, this is the of course is not a non-dimensional I am not d dimensionalize the time yet. Now, if I define theta equal to t u by L t u by L or in other words suppose I multiply this equation by L by u by L by u. So, let us multiply both the sides by L by u then what do I get d A divided by u L I am multiplying it by L by u L by u. So, u L this will go away L by u means what when it comes in the denominator you have this happening. So, L by you multiplied by L by u comes in the denominator 1 upon u by L and I take this here t into u by L. So, I get d theta del theta. So, this is the dimensionless equation theta have got right. If it is not clear you can do it on your own and check. So, dimensionless concentration dimensionless length dimensionless time what is unit of this second meters per second and meter. So, no unit u by L has a unit per second. So, that is why I am using u by L as a parameter here or other I have clubbed this 2 to get dimensionless time and d A by u L what is the unit of d A meter square per second this is meters per second and this is meter. So, this is again dimensionless number. Rest all on dimensionless quantity. So, this is the dimensionless number.  This is a very famous number which this is this number that characterizes the mixing right is named after an engineer called Peclet. But of course, the Peclet number is inverse of this. That means, u L by d A. So, Peclet number is u L by d A or in other words 1 upon p is equal to d A by u L Peclet number that t is silent. So, it is a Peclet number the very famous number. So, let me put that number in the equation again you remember that equation. So, here right. So, d A by u L is 1 divided by p 1 by p right. So, this is my equation. So, this Peclet number before we go ahead and understand the meaning of it. If Peclet number is very large if Peclet number is very large that means the dispersion is small velocity is very large what does it mean? It means I am going towards what kind of flow plug flow because dispersion is less velocity is very large no dispersion or a very small dispersion Peclet number is very large if it is infinity then it is plug flow reactor. In other words if dispersion coefficient is very large if dispersion coefficient is very large then Peclet number will become very small and what is the meaning of large dispersion that means there is so much back mixing happening and what is that situation? It is nothing but a back mix reactor a CSTR when Peclet number is 0. So, 2 bounds see it is quite similar to number of CSTR in series number of tanks in series. So, this is a differential equation our continuous dynamics there it was discrete dynamics theoretically both are falling in line or we have got similar. So, Peclet number can be defined in words as Peclet number is equal to rate of transport by convection see those two terms U L in the or U in the numerator U U is convection is a velocity divided by rate of transport by dispersion because in the denominator you have dA here you have U and of course L that is characteristic length comes because of like you have to somehow dA means not dA is meter square per second. So, the dA by delta has to come. So, you have that L coming in the expression for packing number, but otherwise remember this U will come in the numerator and dA with the denominator and Peclet number will characterize the back mixing larger the Peclet number smaller is the back mixing and smaller the Peclet number larger back mixing Peclet number infinity means plug flow Peclet number 0 means CSTR. So, these were the boundary conditions I had if I non dimensionalize them. So, the first one looks like this the first one if I non dimensionalize this it becomes 1 by Pe into del psi by del lambda plus psi is equal to 1 you can derive it if I non dimensionalize this this is the expression that I get because this is I am just dividing both the sides by this number this number C t 0 sorry C t 0 I am dividing. So, C t 0 by C t 0 1 and then I get this. So, this is the non dimensional form of the boundary condition 1 then the boundary condition 2 this is the boundary condition 2 this was this was boundary condition 1 this is the boundary condition 2. Now, I get it is quite simple d C t by del C t by del z is equal to 0 means del psi by del lambda is equal to 0 fine. So, these two boundary conditions and of course, I will have the initial condition which says that concentration is 0 everywhere at time is equal to 0. So, psi is equal to 0 psi is equal to 0 let me write it psi is equal to 0 at all lambdas when theta what is theta time is equal to 0 I am working in terms of dimensionless numbers. So, with this now I have equation to be solved one equation ordinary sorry partial differential equation second order with three boundary conditions of course, two boundary conditions and one initial condition in dimensionless form what I will get is the plot of plot of tracer concentration say psi versus theta right how will I so what is the tracer concentration at the outlet tracer concentration at the outlet exit concentration because I am not looking at what happens inside of course, it is going to be in between what you get at the outlet and what you get in the what you injected in like. But normally see what do I realize I realize the outlet concentration because I do pulse experiment or I do tracer injection experiment. So, this is the outlet concentration when I solve this equation when I solve this equation at z is equal to l or at lambda is equal to 1 what is the concentration this is what I am going to see right. So, what kind of profile will I see here. So, it all depends on the Peclet number if Peclet number is very very large if Peclet number is very very large means dispersion is 0 right or say Peclet number is infinite dispersion is 0 what is the tracer concentration that I am going to see it is exactly similar to what I have injected I am going to see a plug flow behavior. So, what is that right this is what I am going to see this is what is this this is theta equal to the dimensionless time theta equal to 1 why dimensionless time it is t into u by l u by l or l by u is tau that is residence time. So, this time is tau this time is tau actual time. So, actual time divided by tau that is theta is equal to 1 right. So, this is when Peclet number is infinite or sometimes you will have like they talk in terms of d by u l. So, d by u l will be 0 or this corresponds to d by u l equal to 0 ok. Another extreme Peclet number is 0 famous e curve for another plotting e curve plotting concentration, but it is ok the qualitative behavior is say similar to the e curve and then in between these two see now this plot has to become this means the transition is like this right sorry this will move this peak will normally increase or like the height of so, because it is going to go up right. So, this is a typical behavior that I will see as the value of Peclet number increases. So, here Peclet number is 0 here the Peclet number is probably say some value 5 Peclet number what 100 and so on. So, this is a typical behavior and this is quite similar to what we have seen before before means what for a tangent series model. In tangent series model we looked at how this profile or how this e curve or the tracer concentration changes with respect to time sorry yeah with respect to time or e curve how does it look like with respect to n rather sorry number of tangent series. So, it goes from CSTR to PFR similar thing is happening here that means n in tangent series model is equivalent to Peclet number in the case of dispersion model right both are similar both are similar right. So, once you have the e curve now is other way round once you have the e curve you can calculate a Peclet number. I have drawn this for several values of Peclet number, but if you do laboratory experiment for tracer injection get some e curve I can calculate a corresponding Peclet number from this right and Peclet number will characterize the extent of mixing like the number of tanks in series in the previous model. So, as simple as that so do a tracer experiment get e curve or concentration versus time for the tracer determine the value of Peclet number what next then if you have value of Peclet number then you have to calculate a conversion when the actual reaction is taking place. So, far we just looked at a tracer experiment without any reaction we have characterized the flow patterns in terms of extent of back mixing and that is through the value of Peclet number. Once you get a Peclet number we have we go for the reaction model we go for the reaction model. Now, how to get a value of Peclet number so one can derive of course, one can solve this equation we have to solve this equation numerically by the way it is a differential equation. So, I have not given you analytical expression for this, but you have to solve this equation numerically or analytical expression is also possible it comes in the form of a series mathematical series but then that is not important see the expression is not important how it looks like what is its qualitative behavior is important and once I have the expression or I have the I have the profile or e curve rather how to get a value of Peclet number. So, one can get an approximate value of Peclet number from this. Now, once I have the e curve from this e curve or concentration tracer concentration versus time curve I know that sigma square that is variance divided by mean residence time which is nothing but tau that is our residence time in this particular volume divided volumetric flow rate tau square infinite 0 to infinite t minus tau square Et dt should be noted that Tm is equal to tau is valid only for closed closed system this is something that I know this is definition of sigma variance. Now, from this differential equation I can derive the expression for sigma in terms of Pe like what I did before in the number of tangent series sigma square was equal to sigma square was equal to tau by n or tau sorry tau square by n square. So, here again I am going to get an expression for sigma in terms of Peclet number. So, that expression happens to be sigma square tau m this is nothing but tau is equal to 2 by Pe minus 2 by Pe square 1 minus e raise to minus Pe slightly complicated no need to remember this equation, but then you know there is a relationship between sigma Pe and Tm or other tau and we had a similar relationship before as well between sigma tau and n there was n here that was relatively simpler. Now, this is slightly complicated but of course no need to worry about it earlier relationship was sigma square is equal to tau square divided by n sorry yeah. Now, instead of this relationship I have this relationship here I had n as a parameter now I have Pe as a parameter that is the only difference. So, once I get sigma square from my e curve that is mean variance I get sigma square from mean variance I can get a value of Pe that is Peclet number and once you have Peclet number you have characterized the flow pattern then the next job is to get a conversion for the reaction of interest. So, far no reaction similar to what we did once you get a value of n then for a first order reaction I say that x is equal to 1 minus 1 by 1 plus tau i k raise to n remember. So, this is what I did after getting a value of n now my next job here is after getting a value of Pe I need to get a conversion in terms of Pe. So, that is what we are going to see in the next lecture how to get conversion from Pe alright. Thank you very much.