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HH4 - Theoretical Background To permit heat transfer...

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Theoretical Background To permit heat transfer analysis of heat exchanger, it is necessary to define a mean temperature difference between cold and hot fluids, which change as the fluids pass through the heat exchanger. The following analysis is referred to co-current flow arrangement and the result is also applicable to counter-current arrangement. The rate of heat transfer dq from the hot to the cold fluid through an elemental area dA about the location A is given by: dq=U dA ∆T (1) This heat transfer dq should be equal to the heat given up by the hot fluid flowing from position A to A+dA h ph h dT C m dq - = (2) Similarly for the heat gain by the cold fluid c h dT dT T d - = ) ( (3) The temperature difference between the hot and cold fluid is c h T T T - = (4) Differentiating equation(4) gives c h dT dT T d - = ) ( (5) Combining equations (2),(3) and (5) ) 1 1 ( ) ( pc c pc h C m C m dq T d + - = (6) Substituting equation (1) into (6) leads to dA C m C m U T T d pc c ph h ) 1 1 ( ) ( + - = (7) Integrating equation (6) over the whole heat exchanger gives pc c ph h C m C m T T q 1 1 2 1 + - = (8) Similarly, by integrating equation (7) over the whole heat exchanger leads to t m pc c ph h A U C m C m T T ) 1 1 ( ) ln( 2 1 + = (9) where Um is the mean heat transfer coefficient = A t m UdA T U 0 1 (10) combining new equations (9) and (11) leads to m l m t m t T U A T T T T U A q , 2 1 2 1 ) / ln( = - = (11) m l T ,. is called the log-mean temperature difference(LMTD)
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) / ln( 2 1 2 1 T T T T T lm - = (12) in case of countercurrent floe when m ph C = c m pc C then Δ 0 T l T in such case L’Hopitals rule may be applied to show that L m l T T T = = 0 , (13) if Δ o T is not more than 50% grater than Δ L T the LMTD given by Eq 14 can be approximated by the arithmetic mean temperature difference to within about 1%. It is to be noted that LMTD is always less than the arithmetic mean. It is also to be noted that LMTD has been derived for single-pass heat exchangers. To use LMTD for multipass heat exchangers a correction factor must be applied . Correction factors are commonly given in graphs applying to a particular heat-exchanger geometry. The overall heat transfer coefficient in Eq1 takes into account thermal resistance in the inside and outside films, resistance due to fouling and resistance due to conduction through the tube wall.
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