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824 ENGINEERING THERMODYNAMICS

Cold

Cold

Hot

Annulus

surrounding

the pipe

Pipe

( ) Flow arrangement.a

dA

Area

th 1

th 2

tc 1

tc 2

q

dtc

dth

th

q 2

( ) Temperature distribution.b

dQ

tc

q 1

Hot fluid

Cold fluid

Fig. 15.39. Calculation of LMTD for a counter-flow heat exchanger.

or dθ = – dQ
11
CChc

−

L

N

M

O

Q

P ...(15.57)

Inserting the value of dQ from eqn. (15.55), we get

dθ = – U dA (th – tc)^11

CChc

−

L

N

M

O

Q

P

= – U dA. θ

11

CChc

−

L

N

M

O

Q

P

or dθ
θ
= – U dA
11
CChc
−
L
N

M

O

Q

P

Integrating the above equation from A = 0 to A = A, we get

ln (θ 2 /θ 1 ) = – U. A

11

CChc

−

L

N

M

O

Q

P ...(15.58)

Now, the total heat transfer rate between the two fluids is given by

θ = Ch (–) (–)tt Ctthh cc c 12 = 21 ...(15.59)

or

(^112)
C
tt
h Q
= hh−
...[15.60 (a)]
or
(^121)
C
tt
c Q
= cc−
...[15.60 (b)]