Dr Grant Campbell University of Manchester February 2013 Heat Transfer & Process Integration The ε ε ε-NTU Method of Heat Exchanger Design 1. Introducing the ε ε ε-NTU Method, illustrating its use, and contrasting it with the LMTD method You have learnt previously, by way of introduction, that heat exchanger design revolves around the equation m T UA Q Δ = There are two main types of heat exchanger problem: • the sizing problem, in which we know the required rate of heat transfer, and need to decide what size of heat exchanger (of a particular configuration) is required to deliver that rate of heat transfer, i.e. we know Q , and we need to calculate A; • the rating problem (sometimes called the performance problem), in which we have an existing heat exchanger, of known configuration and size, and we want to calculate what rate of heat transfer it will deliver, i.e. we know A, and we need to calculate Q . In either case, we need to be able to calculate U, the overall heat transfer coefficient, and ΔT m , the effective average temperature driving force within the heat exchanger. The basis of the LMTD method of heat exchanger design is the equation: lm T T UAF Q Δ = where ΔT lm is the log-mean temperature difference. The factor F T indicates how far the flow of the two fluids in the heat exchanger deviates from counterflow behaviour, and hence how much less the effective temperature driving force is than that operating under true counterflow behaviour. The problem is that to calculate ΔT lm requires us to know all four temperatures, i.e. the inlet and outlet temperatures for both the hot and the cold stream. Frequently we only know the inlet temperatures, and want to know what outlet temperatures will be achieved, without having to specify them in advance. This is particularly the case for rating problems, in which we have an existing heat exchanger, and want to know how it will perform if we use it for two fluids of known initial temperatures. In this situation, using the LMTD method would require an iterative solution in order to find combinations of outlet temperatures that give equal heat transfer between the two fluids and across the heat exchanger. The ε-NTU method offers the advantage that only the two inlet temperatures need to be known, and avoids the need for iterative solutions. The starting point for the ε-NTU method, by contrast, is the equation: ( ) ( ) in out p T T c m Q - = For a heat exchanger, the heat lost by the hot stream is equal to the heat gained by the cold stream, which is equal to the heat transfer rate: ( ) ( ) ( ) ( ) 1 2 2 1 c c c p h h h p T T c m T T c m Q - = - =
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Dr Grant Campbell University of Manchester February 2013
Heat Transfer & Process Integration
The εεεε-NTU Method of Heat Exchanger Design
1. Introducing the εεεε-NTU Method, illustrating its use, and contrasting it with the LMTD
method
You have learnt previously, by way of introduction, that heat exchanger design revolves
around the equation
mTUAQ ∆=�
There are two main types of heat exchanger problem:
• the sizing problem, in which we know the required rate of heat transfer, and need to
decide what size of heat exchanger (of a particular configuration) is required to deliver
that rate of heat transfer, i.e. we know Q� , and we need to calculate A;
• the rating problem (sometimes called the performance problem), in which we have an
existing heat exchanger, of known configuration and size, and we want to calculate what
rate of heat transfer it will deliver, i.e. we know A, and we need to calculate Q� .
In either case, we need to be able to calculate U, the overall heat transfer coefficient, and
∆Tm, the effective average temperature driving force within the heat exchanger.
The basis of the LMTD method of heat exchanger design is the equation:
lmT TUAFQ ∆=�
where ∆Tlm is the log-mean temperature difference. The factor FT indicates how far the flow
of the two fluids in the heat exchanger deviates from counterflow behaviour, and hence how
much less the effective temperature driving force is than that operating under true
counterflow behaviour. The problem is that to calculate ∆Tlm requires us to know all four
temperatures, i.e. the inlet and outlet temperatures for both the hot and the cold stream.
Frequently we only know the inlet temperatures, and want to know what outlet temperatures
will be achieved, without having to specify them in advance. This is particularly the case for
rating problems, in which we have an existing heat exchanger, and want to know how it will
perform if we use it for two fluids of known initial temperatures. In this situation, using the
LMTD method would require an iterative solution in order to find combinations of outlet
temperatures that give equal heat transfer between the two fluids and across the heat
exchanger. The ε-NTU method offers the advantage that only the two inlet temperatures
need to be known, and avoids the need for iterative solutions.
The starting point for the ε-NTU method, by contrast, is the equation:
( )( )inoutp TTcmQ −= ��
For a heat exchanger, the heat lost by the hot stream is equal to the heat gained by the cold
stream, which is equal to the heat transfer rate:
( ) ( ) ( ) ( )1221 cccphhhp TTcmTTcmQ −=−= ���
The ε-NTU Method of Heat Exchanger Design 2
Dr Grant Campbell University of Manchester February 2013
Considering the cold fluid, entering at a temperature Tc1, the maximum possible rate of heat
transfer would correspond to the maximum possible temperature rise of the cold fluid. Under
countercurrent flow, the theoretical maximum temperature that the cold fluid could reach is
the inlet temperature of the hot fluid, i.e. if Tc2 = Th1. (This is a theoretical maximum, as it
would require an infinitely large heat exchanger to achieve.) Thus, the maximum theoretical
rate of heat transfer is given by
( ) ( )11max chcp TTcmQ −= ��
T
Th1
Th2
Tc2
Tc1
�H|∆ �H | = �Q
Slope = hPcm )(
1
�
Slope = cPcm )(
1
�
T
Th1
Th2
Tc2
Tc1
�H�Q
maxQ�
The ε-NTU Method of Heat Exchanger Design 3
Dr Grant Campbell University of Manchester February 2013
Equally, the maximum heat transfer rate from the hot fluid would occur if the hot fluid exited
at the temperature of the inlet cold fluid:
( ) ( )11max chhotp TTcmQ −= ��
In practice, one of these will be limiting. The temperature change is greatest for the fluid that
has the smallest capacity to hold heat, i.e. the one with the smaller ( )pcm� . The other fluid
will have a smaller temperature change – it could therefore never reach the inlet temperature
of the first fluid, no matter how large the heat exchanger. Therefore, the maximum
theoretical heat transfer is given by
( ) ( )11minmax chp TTcmQ −= ��
Now, the actual rate of heat transfer is smaller than this theoretical maximum, because the
heat exchanger is not infinitely large. We can define an effectiveness, ε, such that the actual
heat transfer is less than the maximum theoretical heat transfer:
( )( )
( )( )11
12
11
21
max
ferheat trans al theoreticMaximum
achievedfer heat trans Actual
ch
cc
ch
hh
TT
TTor
TT
TT
Q
Q
−
−
−
−==
=
�
ε
depending which stream has the larger temperature change.
Then, the actual heat transfer achieved is given by
( ) ( )11minmax chp TTcmQQ −== ��� εε
Now, we generally know the inlet temperatures of our two fluids – Th1 and Tc1. We also
generally know their respective mass flowrates and specific heat capacities. So, if we can
find a way of calculating the effectiveness, ε, then we can calculate the actual heat transfer.
(Note that ε looks suspiciously like P, one of the parameters used to calculate FT in the
LMTD method. In fact, P is similarly defined as the thermal effectiveness:
( )
( )
( ) ( )
( ) ( )max11
12
11
12
Q
Q
TTcm
TTcm
TT
TTP
chcoldp
cccoldp
ch
cc
�
�
�
�
=−
−=
−
−=
The subtle difference is that P is defined relative to the cold fluid, irrespective of whether it is
the one that could theoretically achieve the greatest temperature change. By contrast, ε is
defined specifically with respect to the fluid that could in theory achieve the greater
temperature change, i.e. the one with the smaller ( )pcm� .)
For different heat exchanger configurations, the effectiveness, ε, can be shown to be a
function of two further dimensionless parameters, C* and NTU:
( )ionconfiguratexchanger heat NTU, ,*Cf=ε
The ε-NTU Method of Heat Exchanger Design 4
Dr Grant Campbell University of Manchester February 2013
The first of these two dimensionless parameters is C*:
max
min*
C
CC =
where ( )minmin pmcC = , the lower of the heat capacity flowrates of the two fluids, and Cmax is
the higher of the two heat capacity flowrates; C* is thus the ratio of these heat capacity
flowrates. Again, this looks spookily familiar; as R in the LMTD method is also defined as
the ratio of the heat capacity flowrates. The subtle difference here is that in calculating R, the
hot fluid heat capacity flowrate forms the denominator, and the cold fluid heat capacity
flowrate is the numerator, irrespective of which is larger, such that R can be greater than 1 –
in fact, R can vary from 0 to ∞ . By contrast, C* is defined such that the smaller of the two
heat capacity flowrates is on the numerator, such that C* can only vary between 0 and 1.
So, for two fluids,
( ) ( )cpchph cmCcmC �� == ;
One of these will be smaller, giving Cmin, while the other one gives Cmax. Note the units of
heat capacity flowrates: K
W
Ks
J
Kkg
J
s
kg==× . This means that C indicates the rate of heat
uptake corresponding to a temperature rise of 1 K.
Now, the other dimensionless parameter that determines the value of ε is NTU, the Number
of Transfer Units. This is defined as
min
NTUC
UA=
Cmin is, as just noted, the smaller of the heat capacity rates of the two streams. The units,
remember, were W/K – the rate at which this stream can take up or lose heat per degree
change in temperature. The product UA also has the units W/K, indicating in this case the
rate of heat transfer per degree of temperature difference – same units, hence NTU is
dimensionless. It represents in dimensionless form the amount of heat transfer available
(which depends on both U and A) to heat a particular heat capacity rate of fluid (where the
heat capacity rate depends on both the flowrate and the heat capacity). So it clearly relates
the size and performance of the heat exchanger to the required amount of heating. A large
value of NTU would imply a lot of heat available to heat a given fluid at a given flowrate.
(Alternatively, NTU can be viewed as the ratio of where the heat has come from, to where the
heat goes. It comes from a temperature driving force between the two fluids, ∆T. It goes into
a temperature rise of the fluid with the smaller heat capacity. NTU therefore represents the
ratio of the temperature rise achieved for every degree difference in temperature driving
force. A high value of NTU will therefore deliver a large temperature rise for a given
temperature driving force, indicating the heat is being transferred effectively – because of a
high heat transfer coefficient and/or a large area.)
Similarly to FT, charts are available that relate ε, NTU and C*. Let’s do an example to
introduce such a chart and see how it’s used. Then we’ll explore some examples that
explicitly derive the relationships between ε, NTU and C*, so that you can see where these
charts some from.
The ε-NTU Method of Heat Exchanger Design 5
Dr Grant Campbell University of Manchester February 2013