Heat Exchangers

Definition of Heat Exchanger: Heat exchangers are device that facilitate the exchange of
heat between two fluids that are at different temperatures while keeping them from mixing
with each other.
Alternatively, a heat exchanger is a device designed to efficiently transfer or "exchange" heat
from one matter to another. When a fluid is used to transfer heat, the fluid could be a liquid, such
as water or oil, or could be moving air.
Application of HX
Types of HX
Overall heat transfer Coefficient
Fouling Factors
Its effect, causes and how to reduce
Analysis of HX
Here, there are two methods available for heat exchangers problems. These methods are as follows:
a. LMTD method (Log mean temperature Difference)
b. Effectiveness NTU method (Number of Transfer Unit)
Before going to these methods we need some considerations. Some assumptions are as follows:
1. Heat exchanger operate for long time, no change in operating conditions i.e. steady flow
conditions,
2. Kinetic energy and potential energy changes are negligible as velocity, mass flow rates,
temperature, elevations are not changes for long time.
3. Specific heats are constant.
4. Axial heat conduction along the tube is usually insignificant and can be considered negligible.
5. No heat loss through the surface of the tube.

The idealizations stated above are closely approximated in practice, and they greatly simplify the
analysis of a heat exchanger with little sacrifice of accuracy. Therefore, they are commonly used.
Under these conditions we have taken First law of thermodynamics that rate of heat transfer from
hot fluid is equal to bethe rate of heat gain by the cold fliud.

Heat gain by cold fliud, Heat loss by hot fluid

Heat capacity rate : Sometimes we find the product of mass flow rate and specific heat of a
fluid is a single quantity called heat capacity rate. For hot and cold fluids heat capacity rates are

The heat capacity rate of a fluid stream represents the rate of heat transfer needed to change the
temperature of the fluid stream by 1°C as it flows through a heat exchanger. Note that in a heat
exchanger, the fluid with a large heat capacity rate will experience a small temperature change,
and the fluid with a small heat capacity rate will experience a large temperature change.
Therefore, doubling the mass flow rate of a fluid while leaving everything else unchanged will
halve the temperature change of that fluid.
Now the heat transfer will be for both hot and cold fluids are:

and

Note that the only time the temperature rise of a cold fluid is equal to the tempera-ture drop of
the hot fluid is when the heat capacity rates of the two fluids are equal to each other.

Fig: Two fluids that have the same mass flow rate and the same specific heat experience the
same temperature change in a well-insulated heat exchanger.
Two special types of heat exchangers commonly used in practice are condensers and boilers.
One of the fluids in a condenser or a boiler undergoes a phase-change process, and the rate of
heat transfer is expressed as Q = mhfg
Here no temperature change, heat capacity rate is infinity. Condenser and boiler temperature
distribution are shown in below.

The rate of heat transfer in a heat exchanger can also be expressed in an analogous manner to
Newton’s law of cooling as Q = UAs ΔTm

U is the overall heat transfer coefficient W/m2.0K. , ΔTm is an appropriate average temperature
difference between the two fluids. As is the heat transfer area.
It turns out that the appropriate form of the mean temperature difference between the two fluids
is logarithmic in nature, and its determination is presented in next section.

LOG MEAN TEMPERATURE DIFFERENCE

The temperature difference between the hot and cold fluids varies along the heat exchanger, and
it is convenient to have a mean temperature difference Tm for use in the relation Q =UAsΔTm In
order to develop a relation for the equivalent average temperature difference between the two
fluids, consider the parallel-flow double-pipe heat exchanger shown in Figure 2, where previous
assumptions are considered. No heat loss through the surroundings, kinetic and potential energy
are constant, specific heats are constant.
Figure 2: Variation of the fluid temperatures in a parallel-flow double-pipe heat exchanger.
Heat loss by hot fluid is 𝑑𝑄 = −𝑚ℎ 𝐶𝑝ℎ 𝑑𝑇ℎ, (1)
Heat gain by cold fluid is 𝑑𝑄 = 𝑚𝑐 𝐶𝑝𝑐 𝑑𝑇𝑐, (2)
𝑑𝑄 𝑑𝑄
From the above equations we find, 𝑑𝑇ℎ = − 𝑚 (3) and 𝑑𝑇𝑐 = (4)
ℎ 𝐶𝑝ℎ 𝑚𝑐 𝐶𝑝𝑐

Taking their difference, Eq. (3)-Eq.(4), we get,

(5)

The rate of heat transfer in the differential section of the heat exchanger can also be expressed as
(6)
𝛿𝑄
≫ 𝑇ℎ − 𝑇𝑐 = (7)
𝑈×𝑑𝐴𝑠
Dividing Eq. (7) by Eq.(5), we find,

(8)

Integrating the Eq.(8) from the inlet of the heat exchanger to its outlet, we obtain,

(9)

Now we can use the Eq. and

In Eq. (9) and thus we have
𝑇ℎ,𝑜𝑢𝑡 −𝑇𝑐,𝑜𝑢𝑡 𝑄 𝑄
𝑙𝑛 = −𝑈𝐴𝑠 (𝑇 +𝑇 ) (10)
𝑇ℎ,𝑖𝑛 −𝑇𝑐,𝑖𝑛 ℎ,𝑖𝑛 −𝑇ℎ,𝑜𝑢𝑡 𝑐,𝑜𝑢𝑡 −𝑇𝑐,𝑖𝑛

Rearranging this we have,

(𝑇ℎ,𝑜𝑢𝑡 −𝑇𝑐,𝑜𝑢𝑡 )−(𝑇ℎ,𝑖𝑛 −𝑇𝑐,𝑖𝑛)
𝑄 = 𝑈𝐴𝑠 𝑇ℎ,𝑜𝑢𝑡−𝑇𝑐,𝑜𝑢𝑡 = 𝑈𝐴𝑠 𝛥𝑇𝑚 (11)
𝑙𝑛
𝑇ℎ,𝑖𝑛 −𝑇𝑐,𝑖𝑛

(𝑇ℎ,𝑜𝑢𝑡 −𝑇𝑐,𝑜𝑢𝑡 )−(𝑇ℎ,𝑖𝑛 −𝑇𝑐,𝑖𝑛)

Where 𝛥𝑇𝑚 = 𝑇ℎ,𝑜𝑢𝑡−𝑇𝑐,𝑜𝑢𝑡 (12)
𝑙𝑛
𝑇ℎ,𝑖𝑛 −𝑇𝑐,𝑖𝑛

𝛥𝑇𝑚 is the log Mean Temperature difference for parallel flow double pipe heat exchanger.
Sometimes this equation can be expressed as (13)

Here ΔT1and ΔT2 are the temperature difference between two fluids at the ends (inlet and outlet).
The arithmetic man ΔTam=1/2(ΔT1+ ΔT2), which is slightly higher than ΔTm. It truly reflects the
exponential decay of the local temperature. Eq. (11) is also for counter flow heat exchanger.
Multipass and cross flow heat Exchangers
Above equations are usable for double pipe single phase problems (parrallel flow and counter
)flow arrangements. Similar relations are also developed for cross-flow and multipass shell-and-
tube heat exchangers, but the resulting expressions are too complicated because of the complex
flow conditions.
In such cases, it is convenient to relate the equivalent temperature difference to the log mean
temperature difference relation for the counter-flow caseas.
∆𝑇𝑙𝑚 = 𝐹 × ∆𝑇𝑙𝑚, 𝐶𝐹 (14)
where F is the correction factor, which depends on the geometry of the heat exchanger and the
inlet and outlet temperatures of the hot and cold fluid streams. The ∆𝑇𝑙𝑚, 𝐶𝐹 is the log mean
temperature difference for the case of a counter-flow heat exchanger with the same inlet and
outlet temperatures.
The correction factor F for common cross-flow and shell-and-tube heat exchanger configurations
is given in Figure 13–18 versus two temperature ratios P and R defined as follows:

(15) and

(16)

Note that the value of P ranges from 0 to 1. The value of R, on the other hand, ranges from 0 to
infinity, with R = 0 corresponding to the phase-change (condensation or boiling) on the shell-
side and R =α to phase-change on the tube side. The correction factor is F = 1 for both of these
limiting cases.
Therefore, the correction factor for a condenser or boiler is F = 1, regardless of the configuration
of the heat exchanger.
Figure 13-18: Correction factor F charts forcommonshell-and-tube and cross-flowheatexchangers
(from Bowman,Mueller,andNagle, Ref. 2).
Solution:
Solution:
Solution:
Which one has higher heat transfer (counter or parallel flow)
For specified inlet and outlet temperatures, the log mean temperature difference for a counter-
flow heat exchanger is always greater than that for a parallel-flow heat exchanger. That is,
∆Tlm,CF › ∆T lm, PF, and thus a smaller surface area (and thus a smaller heat exchanger) is needed
to achieve a specified heat transfer rate in a counter-flow heat exchanger. Therefore, it is
common practice to use counter-flow arrangements in heat exchangers. For same area, counter
flow has higher temperature difference than parallel flow arrangements which causes higher heat
transfer rate. Thus counter flow system is more effective than parallel flow.

In a counter-flow heat exchanger, the temperature difference between the hot and the cold fluids
will remain constant along the heat exchanger when the heat capacity rates of the two fluids are
equal (that is, ∆T = constant when Ch = Cc or mhCph = mcCpc). Then we have In a counter-flow
heat exchanger, the temperature difference between the hot and the cold fluids will remain
constant along the heat exchanger when the heat capacity rates of the two fluids are equal (that
is, ∆T1= constant when Ch = Cc or mhCph = mcCpc. Then we have ∆T1 =∆T2, and the last log
mean temperature difference relation gives ∆Tlm = 0/0 , which is indeterminate. It can be shown
by the application of l’Hôpital’s rule that in this case we have
∆Tlm =∆T1 = ∆T2, as expected.

When does LMTD Applicable?

The log mean temperature difference (LMTD) method is easy to use in heat exchanger analysis
when the inlet and the outlet temperatures of the hot and cold fluids are known or can be
determined from an energy balance. Once ∆Tlm, the mass flow rates, and the overall heat trans-
fer coefficient are available, the heat transfer surface area of the heat exchanger can be
determined from Q = UAs ∆Tlm
Therefore, the LMTD method is very suitable for determining the size ofaheat exchanger to
realize prescribed outlet temperatures when the mass flowrates and the inlet and outlet
temperatures of the hot and cold fluids are specified.
What are the steps taken for solving by LMTD
With the LMTD method, the task is to select a heat exchanger that will meet the prescribed heat
transfer requirements. The procedure to be followed by the selection process is:
1. Select the type of heat exchanger suitable for the application.
2. Determine any unknown inlet or outlet temperature and the heat transfer rate using an energy
balance.
3. Calculate the log mean temperature difference Tlm and the correction factor F, if necessary.
4. Obtain (select or calculate) the value of the overall heat transfer co-efficient U.
5.Calculate the heat transfer surface area As .

When is effectiveness NTU method applicable instead of LMTD method?

When a system has no outlet temperatures (hot and cold fluid) then it is not easy to
find LMTD hence the actual heat transfer rate.
In that case effectiveness NTU method is suitable for determining the outlet
temperatures and actual heat transfer rate as well as the size of the heat exchanger.

For this a new term is introduced as effectiveness of heat exchanger. The method is
called effectiveness NTU method.

Effectiveness NTU method

Definition of effectiveness of HX: It’s the ratio of actual heat transfer rate to the maximum
possible heat transfer rate.
Mathematically, the heat transfer effectiveness ε, defined as

The actual heat transfer rate in a heat exchanger can be determined from an energy
balance on the hot or cold fluids and can be expressed as

where Cc = mcCpc and Ch = mhCph are the heat capacity rates of the cold and the hot
fluids, respectively.

Some problems based on effectiveness NTU method:

EXAMPLE 13–4 Heating Water in a Counter-Flow Heat Exchanger A counter-flow double-pipe
heat exchanger is to heat water from 20°C to 80°C at a rate of 1.2 kg/s. The heating is to be
accomplished by geothermal water available at160°C at a mass flow rate of 2 kg/s. The inner
tube is thin-walled and has a diameter of 1.5 cm. If the overall heat transfer coefficient of the
heat exchanger is 640 W/m2 · °C, determine the length of the heat exchanger required to achieve
the desired heating.
Solutions:
Solution: The schematic diagram is shown in Fig.13-30.
The outlet temperatures are not given. From energy balance
it cannot be found. Using LMTD, it needs tedious iteration
for outlet temperatures. But using effectiveness NTU method
It is easy. At first, we know the heat capacity rates of hot and
cold fluids.