HEAT EXCHANGERS
It is often necessary to transfer heat from one fluid to another
without allowing the two fluids to mix or diffuse into one another.
e.g.
A Car radiator:
Water Þ Air
We therefore need to study the heat transfer from one fluid to
another through a solid boundary.
Because many heat exchangers use tubes to conduct one or
both fluids - we shall study a circular wall.
Heat transfer path
Twi
Thot
Thermal
Resistances
Total thermal
resistance
Heat transfer rate :
HEATXS1A.PPP
Two
r
i
1
ho Ao
ln( ro )
1
hi Ai
q total =
Tcold
2plL
1
hi Ai
+
DT
Q& =
=
q total
r
i
ln( ro )
2 plL
>
+ ho1Ao
Thot - Tcold
1
hi Ai
+
r
i
ln( ro )
2 plL
+ ho1Ao
1
The Overall Heat Transfer Coefficient
If we define:-
Q& = UA(Thot - Tcold )
then U is the Overall Heat Transfer Coefficient
Note that U is similar
. to a surface heat transfer (or film)
coefficient used in Q=hADT for convective heat transfer.
It follows that:-
ln( rr )
1
1
1
= q total =
+
+
UA
hi Ai 2plL ho Ao
o
i
but... which area should we use for A? Ai or Ao
Strictly it doesn’t matter, because whichever area we use, the
value of U ‘adjusts’ depending on which area it is based.
However, by convention, we normally use the hot-side area.
If the tube is thin-walled and metal, Ao @ Ai ; ro @ ri, and l is high,
therefore the middle (ln) term is small compared to the remaining
terms. It follows that:
1 1 1
@ +
U hi ho
HEATXS1A.PPP
2
Heat Exchanger Types
The physical design and shape of heat-exchangers varies
widely depending on the purpose for which it is used; the hot
and cold fluids; and the temperatures and pressures
encountered.
However, three basic heat-exchanger (HX) types may be
recognised:
Parallel-Flow HX
Counter-Flow HX
Cross-Flow HX
unmixed
mixed
HEATXS1A.PPP
3
Heat Exchanger Design
In designing a heat exchanger to heat (or cool) a fluid we would
normally know the following:
The fluid to be heated , i.e. the cold fluid
Its inlet temperature Tc1
m& c
Its mass flow rate
The required outlet temperature Tc2
The fluid used to heat the cold fluid, i.e. the hot fluid
Th1
Its inlet temperature
Its mass flow rate or its exit temperature m
& h or Th 2
Subscript ‘1’ denotes inlet
‘2’ denotes outlet
‘c’ denotes ‘cold’
‘h’ denotes ‘hot’
We normally assume that no heat is lost from the HX to the
surroundings, therefore we may write (assuming sensible heat
fluids):-
m& c cc (Tc 2 - Tc1 ) = UA( DT ) = m& h ch (Th1 - Th 2 )
i.e. the heat transfer rate to the cold fluid equals the heat
transfer rate from the hot fluid. The middle term is the heat
transfer rate across the walls of the HX.
HEATXS1A.PPP
4
Looking first at the fluids:
m& ccc (Tc 2 - Tc1 ) = m& hch (Th1 - Th 2 )
This is known
We need to know or decide on Th1 ,
and specify either m& h or Th 2
Note:
The inlet temperature of the hot fluid (Th1) must be hotter than
the required cold fluid exit temperature (Tc2) !... but how much
hotter?
(a) If Th1 is only slightly hotter than Tc2, the two fluids will have
to be kept in thermal contact for a long time to ensure that the
required temperature is reached - this implies a large area of
heat-exchange surface - i.e. an expensive HX.
(b) If Th1 is very much hotter than Tc2 , then only a ‘brief
encounter’ is required - this implies a small area of heatexchange surface (i.e. a cheap HX) but the hot fluid will leave
very much hotter than it otherwise might have, i.e. energy may
be being wasted.
The temperature we specify is therefore a compromise between
HX cost and energy considerations. Methods are available for
optimising the compromise on a life-time cost basis using
Second Law analyses.
HEATXS1A.PPP
5
Mass flow considerations
For a given hot fluid inlet temperature we can plot the variation
in exit temperature with mass flow as shown below:
(Assuming a parallel type HX)
.
Th1
Increasing
Th2
m& h
Tc2
Tc1
Distance through the HX
.
Note:
& h large, Th2 approaches Th1 , and the temperature
With m
differences between the two fluids all along the path are also
large - i.e. a smaller HX surface area will be required - cheap,
but higher pumping power may be required, and energy may be
wasted.
& h required given by:
For a given Th1 there is a minimum m
Th2 = Tc2
However, this implies infinite HX surface area - expensive (!),
but lower pumping power would be required for the lower mass
flow. Again a compromise is needed between capital cost and
running cost over the system’s lifetime
Having decided on temperatures & mass flows, how do we
design a HX with sufficient surface area to achieve our
specifications?
HEATXS1A.PPP
6
In order to design a HX (find the required surface area) using:
Q& = UA ( D T )
we need to find a representative temperature difference, and
find the Overall Heat Transfer Coefficient for the flow conditions
Representative temperature difference
Assume a parallel flow HX:
Th1
Th2
Tc2
Tc1
Distance through the HX
At any station through the HX the temperature between the hot
and cold streams differs, and therefore the heat transfer rate
differs. We need to find the overall effect by integrating along
the path length of the HX.
dT
(negative)
Th
^
^
Tc
^ (Th - Tc)2
^ dT
(Th - Tc)1
^
^
dA
2
1
We can find the change in temperature difference:
(Th - Tc )1 = (Th - Tc )2 + d Tc - d Th
\ ( Th - Tc ) 2 - ( Th - Tc )1 = - d Tc + d Th
or
- d ( Th - Tc ) = dTc - dTh
HEATXS1A.PPP
(i)
7
A heat balance
must also
occur
dQ& = m& c ccdTc = - m& h chdTh = UdA(Th - Tc )
\ dTc =
UdA(Th - Tc )
UdA(Th - Tc )
& dTh =
m& c cc
m& h ch
i.e. dTc =
Substituting
in (i) gives:
-d (Th - Tc ) =
\-
Integrating
from inlet to
outlet gives:
but
(Th 2 - Tc 2 ) æ 1
1 ö
=ç
+
÷ UA
(Th1 - Tc1 ) è m& c cc m& h ch ø
- ln
UA (Tc 2 - Tc1 )
UA (Th1 - Th 2 )
=
&
=
m& c cc
( DT )
m& h ch
( DT )
(Th 2 - Tc 2 ) æ (Th1 - Th 2 ) + (Tc 2 - Tc1 ) ö
=ç
÷
( DT )
ø
(Th1 - Tc1 ) è
(Th1 - Tc1 ) æ (Th1 - Tc1 ) - (Th 2 - Tc 2 ) ö
=ç
÷
( DT )
ø
(Th2 - Tc2 ) è
( DT ) =
(Th1 - Tc1 ) - (Th2 - Tc2 )
ln
HEATXS1A.PPP
(iii)
Q& = UA( DT ) = m& c cc (Tc 2 - Tc1 ) = m& h ch (Th1 - Th 2 )
or ln
therefore:
U (Th - Tc )dA U (Th - Tc )dA
+
m& c cc
m& h ch
d (Th - Tc ) æ 1
1 ö
=ç
+
÷ UdA
(Th - Tc ) è m& c cc m& h ch ø
- ln
\
Substituing in
(iii) gives:
U (Th - Tc )dA
U (Th - Tc )dA (ii)
& dTh =
m& c cc
m& h ch
(Th1 - Tc1 )
(Th2 - Tc2 )
8
(DT) is the LOG MEAN TEMPERATURE DIFFERENCE
It applies to both parallel and counter-flow HX’s
It may be written:
DTlog =
DTin - DTout
æ DTin ö
lnç
÷
D
T
è out ø
It is important to note that the subscripts ‘in’ and ‘out’ should be
tied to either the hot fluid or the cold fluid.
The Overall Heat Transfer Coefficient
In order to find U, we need to find the surface heat transfer
coefficients from a knowledge of the flow conditions inside the
HX. This presumes we know the geometry (otherwise we could
not find Re etc.). Clearly we have to make some initial
assumptions, and then proceed by trial and error toward a final
design. Having found hi and ho we can use the equations on
page 2 to find U.
Finally, having found or estimated DTlog and U we can find A
from:
Q& = UADTlog
To a large extent, the required surface area determines the
size and cost of the HX.
HEATXS1A.PPP
9
Heat Exchangers other than Parallel or Counter Flow
In order to make a heat-exchanger more compact, or to suit a
particular application, it may not be possible (or desirable) to
use a straight parallel or counter flow heat-exchanger.
It is possible to allow one fluid to ‘pass’ the other fluid more
than once.
One shell-pass two tube-pass design.
It is clear that the heat-exchanger is operating under both
parallel and counter flow conditions.
In order to account for these, and other composite designs, it
is necessary to correct the LMTD. This is done using
correction factor tables.
Q& = UAF D Tlog
F, the correction factor, is a function of the design, the
Thermal Capacity Ratio, C and the Effectiveness, E.
NB On the following charts: R = C, and P = E
HEATXS1A.PPP
10
HEATXS1A.PPP
11