TITLE.PM5

dharm
\M-therm\Th15-3.pm5

HEAT TRANSFER 821

Total heat transfer rate in the heat exchanger, Q = UA θm ...(15.48)

Hot fluid

Cold fluid

Heat exchanger

Heat transfer area

θ 1

th 1

tc 1

Q

Q

θ 2

th 2

tc 2

Fig. 15.37. Overall energy balance in a heat exchanger.

where, U = Overall heat transfer coefficient between the two fluids,
A = Effective heat transfer area, and
θm = Appropriate mean value of temperature difference or logarithmic mean temperature
difference (LMTD).
15.4.4. Logarithmic Mean Temperature Difference (LMTD)
Logarithmic mean temperature difference (LMTD) is defined as that temperature difference
which, if constant, would give the same rate of heat transfer as actually occurs under variable condi-
tions of temperature difference.
In order to derive expression for LMTD for various types of heat exchangers, the following
assumptions are made :

1. The overall heat transfer coefficient U is constant.


2. The flow conditions are steady.


3. The specific heats and mass flow rates of both fluids are constant.


4. There is no loss of heat to the surroundings, due to the heat exchanger being perfectly
   insulated.


5. There is no change of phase either of the fluids during the heat transfer.


6. The changes in potential and kinetic energies are negligible.


7. Axial conduction along the tubes of the heat exchanger is negligible.
   15.4.4.1. Logarithmic Mean Temperature Difference for “Parallel-flow”
   Refer Fig. 15.38, which shows the flow arrangement and distribution of temperature in a
   single-pass parallel-flow heat exchanger.
   Let us consider an elementary area dA of the heat exchanger. The rate of flow of heat through
   this elementary area is given by
   dQ = U dA (th – tc) = U. dA. ∆t
   As a result of heat transfer dQ through the area dA, the hot fluid is cooled by dh whereas the
   cold fluid is heated up by dtc. The energy balance over a differential area dA may be written as
   dQ = – m&h. cph. dth = m&c. cpc. dtc = U. dA. (th – tc) ...(15.49)
   (Here dth is – ve and dtc is + ve)

or dth = –
dQ
mc

dQ

&hphhC

=–