 I am Ganesh Biyagalai working as an assistant professor in Department of Mechanical Engineering Vulture Institute of Technology. In this lecture of heat exchangers we will study LMTD approach of heat exchanger analysis learning outcome. At the end of this session students will be able to describe overall heat transfer coefficient and select heat exchanger by using analysis LMTD or and NTU approach. Here before starting we must know log mean temperature difference can be denoted by capital letters LMTD. To use the LMTD we must know all inlet and outlet temperatures must be known to us then and then only we can adopt the LMTD approach. There is one concept known as overall heat transfer coefficient and it is usually denoted by capital U. Here the example is taken in which this is the inner tube and this is the outer tube means this becomes concentric tube heat exchanger. Now the transfer of heat through the inner tube takes place by means of convection conduction and convection to the annular fluid. This is the enlarged view of this section. So this is the inner surface of the inner tube outer surface of the inner tube and the heat transfer will flow from hot fluid to the cold fluid. Suppose T i is the inlet fluid temperature of hot fluid, T o is the outlet temperature of the cold fluid then first resistance will be convective resistance, second resistance will be conductive resistance of the wall and this will be the convective resistance on the outer surface. A i is the inner surface area, A o is the outer surface area. H i represents inside overall heat transfer sorry inside heat transfer coefficient and H o outside heat transfer coefficient. Now rate of heat flow is nothing but temperature difference to the total thermal resistance. Now how much is the rate of heat transfer q and resistances are 1 2 3 means convection conductive and convective resistances. So those resistances are 1 by H into A based on inner side plus delta x is the thickness of the wall or tube by K a plus 1 by H into A of convective resistance on outer side. Usually q can be written as UA delta T overall. Now if we compare equation this equation first equation and second equation then I can write u is nothing but 1 by these many resistances u is equal to 1 by first resistance plus second resistance plus first third resistance. So nothing but this is the overall resistance or total resistance. So overall heat transfer coefficient is the reciprocal of total resistance. Here we have the typical values of overall heat transfer coefficient for the fluid combination water to water 850 to 1700 for water to oil 110 to 350. So in this fashion we have the readymade chart of the typical values of overall heat transfer coefficient. Now with this we will start the analysis of heat exchanger. In this analysis there are two methods first method is LMTD method, second method is NTU method. Initially we will go to the LMTD method. Now in the analysis the rate of heat lost by hot fluid you think over it whether it will be equal to cold fluid or we will have some difference think over this obviously the total rate of heat transfer from hot fluid to cold fluid will be same. The heat rejected by hot fluid mcp delta T will be equal to cold fluid heat gain mcp delta T. Now we will adopt LMTD approach for parallel flow heat exchanger. Here this is the hot fluid, this is the cold fluid, hot fluid inlet temperature, hot fluid outlet temperature, cold fluid inlet temperature, cold fluid outlet temperature. The type of the flow is parallel flow and this is the temperature profile for hot fluid and cold fluid. So let delta T represents the inlet fluid temperature difference delta Te exit temperature difference. We can write the equation dq is equal to u area is b into dx and hot fluid temperature minus cold fluid temperature. Rate of heat transfer is mcp delta T which type of the heat transfer process is taking place here is it the sensible heat or latent heat yes it is a sensible heat. So only one relation will be used that is mcp delta T. Now why minus sign is there for the hot fluid because the slope is negative. So now delta T as we have seen Th minus Tc if I differentiate this equation I can write d delta T is equal to d Th minus d Tc. Since dT by delta T is equal to minus 1 by mh into cph plus m by mccpc. So this is the heat flow rate of the hot fluid, heat flow rate of the cold fluid into eub. Now I can cover the total area total surface of the heat exchanger by taking integration that is delta i to delta e similarly here taking integration 0 to l of dx. Then I can write the equation ln del T by del T is equal to minus 1 by mhcph plus 1 by mccpc into ua. So this equation in this equation delta Te is equal to delta Th e minus Tc and delta Ti is equal to Thi minus Tci. Then Q is the total rate of heat transfer in the heat exchanger then I can have the rate of heat transfer same for the cold fluid and hot fluid then 1 by mc can be replaced by 1 by Q this temperature difference for hot fluid similarly I can take it for cold fluid. Then I can write Q is equal to ua this is delta Ti minus delta Te as we have seen delta Ti means inlet temperatures or difference of those fluids that is Thi minus Tci and delta Te is means Th e minus Tce. So I can obtain the equation of rate of heat transfer equal to ua delta Ti minus delta Te divided by ln delta Ti by delta Te. Now here if I will consider this ratio which is which has the log which has the log this log that is why I can use the notation either delta Tm means log mean temperature difference or Elm Td. So for parallel flow heat exchangers the log mean temperature difference is the temperature difference of the fluid set inlet minus temperature difference of the fluid set outlet to the log of delta Ti to delta Te. For further study you can refer fundamentals of heat and mass transfer by Incropera David. Thank you.