Plate Fin Heat Exchanger: Analysis
Welcome to this lecture, in this lecture on plate fin heat exchanger we are going to talk about the analysis of plate fin type heat exchanger. And in this one we will first start with the fin equation where you might have noticed that we have so far in case of plate fin type exchangers we have talked about the separating plates. And in between the plate what we have is basically some fin connected between this one. Between the two separating layers we have since joined and we have talked about the bridging between this plate and the separating I’m in separating plate and the field and we have ensured that there is a good thermal contact between the plate and the thing so you know if we look intowe can think it as a friend connected between the two separating plate so what we need to analyse finally or what is eventually comes as this is some plate at the temperature in a and some plate the AC temperature be depending on the type of fluid and we have like two plates connected by a separating separating I’m in this is the thing which is connected between the two plates so we need to analyse this one or holiday if you have analysed this kind of innovation well most of the time we see that the snow we have a plate and we have a fin and the one end of the phone is at it articular either at that particular temperature or we often considered thatand it is at Amberley communicating with some fluid and there is some e-transfer age between 20 and the surrounding you also often use that this chain is also at a particular temperature Assam temperature at infinity or some other temperature so this is typically the base and the phone is connected to the base and this is how it is looking like in contrast to this one we have another fix the flat and we have enough designated as he and the TV to different temperatures so similar look at the complete assembly we will find that there are different type of layers of things and I mean this is like this this is designated for fluid one and this is fluid to this is a game to 81 and 32 and so on but eventually if you look at it will finally be solving at anytime and I will say there will be the city wall 13 wall-to-wall 3 and 4 115 Walpole and 215 like that it will continue in the actual exchanges but for a particular there I’m in for this particular hair on one side we had 31 and this other side we have I mean I mean between suite 2A minifor consider this 32 on one side it was driving through it one on the other side it is Irving Street they want and what the size it is 7:21 similarly if he considered this Friday on either side of it is having the flu to this is particularly the caseI like that we have and they are challenged so basically it boils down to a situation where we have to separating plate and when connecting between the two have to have to analyse thisthis side is this is the pain and this phone is connected to a temperature here and the side is connected to temperature TV know if we are trying to solve that Finn equation we what we do is that we take a small section of this one and it’s actually a three-dimensional this dimension we’re not drawing this is how it looks like this and this is how it looks like we have taken a small section of them up this friend in this is the x-direction and this is between x and x + BX so within the small element of this thing we have considered a small element DX between x and x + 3 x so we have q x amount of heat conduction heat getting transferred and this is going through this one this element 2x + x play and in-between you knowwhatever shoot that is coming to the film The contracted fluid flowing on top of it on top of this thing is taking that amount of itso hard to make an energy balance what we find is that q x amount of heat is getting distributed 2QX plastics and we have also been convicted down now if we look into this QX + guess you will find that q x + 3 x x + QX + BX we can write it as QX + d qxd xn2x soap we have considered a small element delta x so we have this is how it is getting pissed about it so we have already talked about QX is equals to QX + BX + q convective now from here’s what we are getting is q x + 3 x – 2 x cube +this will become dqxq x x x with the negative term and that will become plus I’m in queue x + 2x – this will become q-connect give is equals to chilltext from here if we look back QX + 3x – 2 x is equals to minus q convective so that’s what we have written here to x + 2 x is equals to minus convective and then we have the builder QTX part this this can be written as CQDX we can write it sorry this the DQ qxt x is equals to we can write it as ddx of minus k into a into dttx so this is what is the convective heat transfer and if we put it in this equation what will find is that Kim convective town is in that queue convective class j2 ttx square with k&a these are the two times we will have for this is equals to zero so if we have this dance what is that in equation we are looking at this is for this cross-section we have felt the heat is 20 weeks and this is u x + 3 x and this is where we have q convective so how much is the q convective heat transfer q convective is basically age into the area what is that area and h a t minus infinity infinity is the fruit temperature and this area is basically nothing but the perimeter x the stairs so here also we have that the expert know if we put into this situation we will find that this is giving you teeth to TTS a square into ka and then you have the text wrong and then we had the we have that into a into 3 – 2 x infinity and that way we can write it in terms of the perimeter x 30 x so this will make the total equation di2 TT services DSS Square is equals to 8 upon p by this guy into a this is nothing but that cross-sectional area through which the conduction it was taking place so that’s about if we talk about this direction as w then we have the blue plus this is the thickness so we have the blue into ex-senators DXL coming out supposed to chillso what’s about the sign now we have that this is the calls to if we put this perimeter be if we put this perimeter p w plus fitness x 2 and if we arrange it here there even find that h in tupi is nothing but two and then we have w + 3 / w in to stay and of course we had that ke part here so there has to be a key here and should have a help now in orphan we make an assumption that the thickness of the film is very I’m in small as compared to the DSW so if we call it tin fin approximation cleanfund approximation we call it approximation so in that case with neck lift as compared to w so in that case we will have twice HW by KWS AT&t service wnw will cancelso this way Khalid SD to t d x square minus m square into 3 minus infinity that is equals to chill if we look into this equation or we have a t2 tea and we put a deliberately this is a constant the great temperature going over the same way as you need to be constantly have this is minus m square and t minus infinity is equal to zero where is m square is nothing but 28 by Katy in general it is HP by k a r 14 fin fin fin assumption this is Naruto but how many MB comes with over 20 by K-Ci this situation shortly write it to BD23 today x squared is equals to m squared theta and this is the same equation that this thing is subjected to convective fluid temperature coming dissipating heat through the contractor will transfer know if we have to solve this equation we know the solution of this equation in general equation theta x what is theatre theatre has been considered to be t minus infinity city is basically the temperature at any location in that feel this is AT&t a and this is that TV anywhere between this one at a location x we consider this temperature to beat ex so this is the temperature at x minus infinity so that the Skeeter a we call it as a minus infinity and theta be it would be TV minus the infinity so now we’re trying to look for solution of this equation and we know what is the solution the theta x I mean theatre basically is equals to some constant intoboundary conditions appropriate for this equation and from there we had to get this constant c1 and c2 and already have learnt how to do that in your earlier classes probably where we have assumed that one of the pain is at constant temperature at the other end is at a diabetic conditions in this case where we have the at x equals to zero we considered it to be theta equals to as we have said Peter a and at x equals 2 l we have three titles to Peter be this is what we have assumed that between the two and the temperature is t&t be so if we solve this equation we will find out I mean if we apply this boundary conditions will find that someone comes to steal one comes to me see one comes to the theatre a * 1 – Omega and c2 becomes theta x Omega this is nothing but is it withbetween the two I’m independent so this is the paint k20a and KB and we have this Omega is equals to e to the power x – hard by to sign hyperbole Carol and c1 and c2 are like this these are the constants they had evaluated for this equation theatre ex-soap the overall equation and what is RR is nothing but a ratio between theta by Peter is so there are other forms of expression and equation of this is this particular expression of finding a solution of an equation solution we have taken from BSC question paper solved know if we look at we have the complete salon sin theta x is equals to thetaplus or minus Alex this is what is the complete solution know if we have the knowledge about the thing temperature profile over the fence so this is what we have the PA and PB and we have the knowledge about the field temperature profile Peter x know and this is a personal pa we call it theatre and theatre be so this is that and this isn’t a bit and we know now the temperature profile over this thing so once we had the temperature profile onto this thing we can try to estimate what is the total amountdo that once we find the total amount of heat getting dissipated over this pain that is 2 into 8 into theta x into DX and this isn’t regarded over the length 02 well why these two factor is coming because we have the surface we have the surface and over the surface we have the heat transfer coefficient age and what is the tax that is basically is the temperature TX liye nuske infinity so that the x minus infinity heat transfer coefficient and the DX so that'sh by MN to Peter a hospital bill if you put that value of theta x and the new intricate cable have an expression of 10 hyperbolic ml by 2 so now what we do is that we divide and multiply both sides by email by 2 so that way what we will get is Emily by two on the side and then we have to hbm then you have to tie a plastic tub is this the thermostat and 10 hyperbolic ml by 2 and here we have multiplied it and divided it but this time hyperbolic MLB mlr this is not thing but the perfect efficiency so we can write it as this to aunt this and they are going out so you have a in2l then you have Peter 8 + theta b and then we write it as ETA this either as a is for alibi to so we call it feel efficiency of the half of the pain so now if we look at we have the heat transfer the total amount of heat transfer that cutie the total amount of heat getting dissipated through the fee is h in 12 and then you have to tie a plastic tub y and x ETA half half in efficiency know if you carefully look into it we can write as if this is Asian too well and then theta and then ate half and so this is one part and the other part if you look at it is just nothing but ancient too well and then theta b and then it a half so as if we have this contribution of half of the fee this is and then half of it is contributing with Peter and this is where we have the theatre be so this is just nothing but a b minus infinity this theatre be and this theatre is nothing but a minus infinity so as if we have half of the fin contributing to the fridge and the other half is connected to the other field so this is basically nothing but the half pinoy realisation of the play funky text near so this is again this is I’m in a good assumption particularly for a two string heat exchangers but when we have multiple streams are we have a different asymmetric when we find that there is a good deal of violation of this one and we will then I know we need to look into a different analysis in during that time so the total heat transfered is as a half of that fin connected to surface A and the other one is connected to Tb. Thank you
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