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CHAPTER 11

HEAT EXCHANGERS

Prepared by: Ibrahim Sezai

ObjectivesWhen you finish studying this chapter, you should be able to:• Recognize numerous types of heat exchangers, and classify

them,• Develop an awareness of fouling on surfaces, and determine

the overall heat transfer coefficient for a heat exchanger,• Perform a general energy analysis on heat exchangers,• Obtain a relation for the logarithmic mean temperature

difference for use in the LMTD method, and modify it for different types of heat exchangers using the correction factor,

• Develop relations for effectiveness, and analyze heat exchangers when outlet temperatures are not known using the effectiveness-NTU method,

• Know the primary considerations in the selection of heat exchangers.

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Types of Heat Exchangers• Different heat transfer applications

require different types of hardware and different configurations of heat transfer equipment.

What should we choose?

Double-Pipe Heat Exchangers• The simplest type of heat exchanger is called the

double-pipe heat exchanger.• One fluid flows through the smaller pipe while

the other fluid flows through the annular space between the two pipes.

• Two types of flow arrangement– parallel flow,– counter flow.

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Compact Heat Exchanger• Large heat transfer surface area per unit volume.• Area density β ─ heat transfer surface of a heat

exchanger to volume ratio.• Compact heat exchanger β >700 m2/m3.• Examples:

– car radiators (β ≈1000 m2/m3),– glass-ceramic gas turbine heat

exchangers (β ≈6000 m2/m3), – the regenerator of a Stirling

engine (β ≈15,000 m2/m3), and – the human lung (β ≈20,000 m2/m3).

• Compact heat exchangers are commonly used in – gas-to-gas and – gas-to liquid (or liquid-to-gas) heat exchangers.

• Typically cross-flow configuration ─ the two fluids move perpendicular to each other.

• The cross-flow is further classified as – unmixed flow

and – mixed flow.

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Shell-and-Tube Heat Exchanger• The most common type of heat exchanger in industrial

applications.• Large number of tubes are packed in a shell with their axes

parallel to that of the shell.• The other fluid flows outside the tubes through the shell.• Baffles are commonly placed in the shell.• Shell-and-tube heat exchangers are relatively large size and

weight.• Shell-and-tube heat

exchangers are further classified according to the number of shell and tube passes involved.

Figure 13.5 Multipassflow arrangements in shell-and-tube heat exchangers.

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Plate and Frame Heat Exchanger

• Consists of a series of plates with corrugated flat flow passages.

• The hot and cold fluids flow in alternate passages• Well suited for liquid-to-liquid heat exchange

applications, provided that the hot and cold fluid streams are at about the same pressure.

The Overall Heat Transfer Coefficient • A heat exchanger typically involves two flowing

fluids separated by a solid wall. • Heat is transferred

– from the hot fluid to the wall by convection,

– through the wall by conduction, and – from the wall to the cold fluid by

convection.• The thermal resistance network

– two convection and – one conduction resistances.

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• For a double-pipe heat exchanger, the thermal resistanceof the tube wall is

• The total thermal resistance

• When one fluid flows inside a circular tube and the other outside of it, we have

( )0ln2

iwall

D DR

kLπ= (11-1)

( )0ln1 12

itotal i wall o

i i o o

D DR R R R

h A kL h Aπ= + + = + + (11-2)

; i i o oA D L A D Lπ π= =

• It is convenient to combine all the thermal resistances in the path of heat flow from the hot fluid to the cold one into a single resistance R

U is the overall heat transfer coefficient, whose unit is W/m2ºC.

• Canceling T, Eq. 11–3 reduces to

i i o oTQ UA T U A T U A T

RΔ

= = Δ = Δ = Δ (11-3)

1 1 1 1 1wall

s i i o o i i o o

R RUA U A U A h A h A

= = = = + + (11-4)

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• When the wall thickness of the tube is small and the thermal conductivity of the tube material is high (Rwall=0) and the inner and outer surfaces of the tube are almost identical (Ai≈Ao≈As), Eq. 11–4 simplifies to

• When hi>>ho

• When hi<<ho

1 1 1

i oU h h≈ + (11-5)

1 1

oU h≈

1 1

iU h≈

Fouling Factor• The performance of heat exchangers usually

deteriorates with time as a result of accumulation of deposits on heat transfer surfaces.

• The layer of deposits represents additional resistanceto heat transfer and causes the rate of heat transfer in a heat exchanger to decrease.

• The fouling factor Rf─ The net effect of these accumulations on heat transfer.

• Two common type of fouling:– precipitation of solid deposits in a

fluid on the heat transfer surfaces.– corrosion and other chemical fouling.

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• The overall heat transfer coefficient needs to be modified to account for the effects of fouling on both the inner and the outer surfaces of the tube.

• For an unfinned shell-and-tube heat exchanger, it can be expressed as

Rf,i and Rf,o are the fouling factors at those surfaces.

( ), ,0ln1 12

f i f oi

i i i o o o

R RD DR

h A A kL A h Aπ= + + + + (11-8)

Analysis of Heat Exchangers• Two different design tasks:1) Specified:

- the temperature change in a fluid stream, and - the mass flow rate.

Required:- the designer needs to select a heat exchanger.

2) Specified:- the heat exchanger type and size,- fluid mass flow rate,- inlet temperatures.

Required:- the designer needs to predict the outlet temperatures and heat transfer

rate. • Two methods used in the analysis of heat exchangers:

– the log mean temperature difference (or LMTD)• best suited for the #1,

– the effectiveness–NTU method• best suited for task #2.

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• The analysis of heat exchangers can be greatly simplified by making the following assumptions, which are closely approximated in practice:– steady-flow,– kinetic and potential energy changes are negligible,– the specific heat of a fluid is constant,– axial heat conduction along the tube is negligible,– the outer surface of the heat exchanger is perfectly

insulated.• The first law of thermodynamics requires that

the rate of heat transfer from the hot fluid be equal to the rate of heat transfer to the cold one.

• The transfer rate to the cold fluid:

• The transfer rate to the hot fluid:

• 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

( ) ( ), , , , ; c pc c out c in c c out c in c c pcQ m c T T C T T C m c= − = − =(11-9) (11-12)

( ) ( ), , , , ; h ph h in h out h h in h out h h phQ m c T T C T T C m c= − = − =(11-10) (11-13)

(11-11)(11-11)

(11-11)

fgQ mh= (11-14)

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Figure 11-13 Variation of fluid temperatures in a heat exchanger when one of the fluid condenses or boils.

Temperature is constant during a phase change (Fig. 11.13)

Since and ΔT = 0,TCmQ pΔ=

then ∞== CCm p

For condensing and boiling: C=∞

For a heat exchanger:

mTUAQ Δ= (11-15)

The Log Mean Temperature Difference Method

• The temperature difference between the hot and cold fluids varies along the heat exchanger.

it is convenient to have a mean temperature difference Tm for use in the relation

• Consider the parallel-flow double-pipe heat exchanger.

s mQ UA T= Δ (11-15)

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• An energy balance on each fluid in a differential section of the heat exchanger

• Taking their difference, we get

h ph h

c pc c

Q m c dT

Q m c dT

δ

δ

⎧ = −⎪⎨

=⎪⎩

(11-16)

(11-17)

hh ph

cc pc

QdTm c

QdTm c

δ

δ

⎧= −⎪

⎪⎨⎪ =⎪⎩

(11-18)

(11-19)

( ) 1 1h c h c

h ph c pc

dT dT d T T Qm c m c

δ⎛ ⎞

− = − = − +⎜ ⎟⎜ ⎟⎝ ⎠

(11-20)

• The rate of heat transfer in the differential section of the heat exchanger can also be expressed as

• Substituting this equation into Eq. 11–20 and rearranging give

• Integrating from the inlet of the heat exchanger to its outlet, we obtain

( )h c sQ U T T dAδ = − (11-21)

( ) 1 1h cs

h c h ph c pc

d T TUdA

T T m c m c⎛ ⎞−

= − +⎜ ⎟⎜ ⎟− ⎝ ⎠(11-22)

, ,

, ,

1 1ln h out c outs

h in c in h ph c pc

T TUA

T T m c m c⎛ ⎞−

= − +⎜ ⎟⎜ ⎟− ⎝ ⎠(11-23)

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• Solving Eqs. 11–9 and 11–10 for mccpc and mhcph and substituting into Eq. 11–23 give

• ΔTlm is the log mean temperature difference.• ΔT1 and ΔT2 are the temperature difference between

the two fluids at the two ends (inlet and outlet).

• It makes no difference which end of the heat exchanger is designated as the inlet or the outlet.

( )1 2

1 2ln

s lm

lm

Q UA TT TT

T T

= Δ

Δ −ΔΔ =

Δ Δ

(11-24)

(11-25)

Counter-Flow Heat Exchangers• The relation already given for the log

mean temperature difference for parallel-flow heat exchanger can be used for a counter-flow heat exchanger.

• ΔT1 and ΔT2 are expressed as shown in the Fig. 11–15.

• ΔTlm, CF > ΔTlm, PF• A smaller surface area (a smaller heat

exchanger) is needed to achieve a specified heat transfer rate in a counter-flow heat exchanger.

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ΔT is constant when Ch = Cc

→ ΔT1=ΔT2 → ΔTlm= indeterminate00

Using l’Hopital’s rule:

→ ΔTlm =ΔT1=ΔT2

A condenser or a boiler can be considered to be either a parallel- or counter-flow heat exchanger since both approaches give the same result.

Multipass and Cross-Flow Heat Exchangers: Use of a Correction

Factor• The log mean temperature difference relation

developed earlier is limited to parallel-flow and counter-flow heat exchangers only.

• To simplify the analysis of cross-flow and multipassshell-and-tube heat exchangers, it is convenient to express the log mean temperature difference relation as

• F is the correction factor, and ΔTlm, CF is the log mean temperature for counter-flow case.

,lm lm CFT F TΔ = Δ (11-26)

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Correction factor F charts common shell-and-tube heat exchangers.

Correction factor F charts: cross-flow heat exchangers.

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The Heat Exchanger Design Procedure using the LMTD

• With the LMTD method, the task is to select a heat exchangerthat 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 ΔTlm 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 needed to meet

requirements.

THE EFFECTIVENESS-NTU METHOD

- The LMTD method is easy to use if the inlet and outlet temperatures are known or can be determined from

( )TCmQ p Δ=

- LMTD method is very suitable for determining the size of a heat exchanger

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If outlet temperatures are not known, then LMTD method is difficult to useIn that case, we use effectiveness-NTU method.

Effectiveness is defined as

ratefer heat trans possible Maximumratefer heat trans Actual

max

==Q

Qε (11-29)

Actual heat transfer rate is

( ) ( )outhinhhincoutcc TTCTTCQ ,,,, −=−= (11-30)

where pccc CmC = and phhh CmC =

In a heat exchanger maximum ΔT between the fluids is

incinh TTT ,,max −=Δ (11-31)

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The heat transfer in a heat exchanger will reach its maximum value when:(1) The cold fluid is heated to the inlet temperature of the hot fluid

(2) The hot fluid is cooled to the inlet temperature of the cold fluid

When Cc ≠ Ch the fluid with the smaller heat capacity rate will experience a larger temperature change,

and thus it will be the first to experience the maximum temperature,

at which point the heat transfer will come to halt.

Therefore, the maximum possible heat transfer rate in a heat exchanger is (Fig. 11.23)

( )incinh TTCQ ,,minmax −=

where Cmin is the smaller of

phhh CmC = and pccc CmC =

Once ε is known, then actual heat transfer rate is determined from

)( ,,minmax incinh TTCQQ −== εεFig. 11-23

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Consider a double-pipe parallel flow heat exchanger

Equation 11-23 developed in the previous section for a parallel-flow heat exchanger can be rearranged as

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

−−

h

c

cincinh

outcouth

CC

CUA

TTTT

1ln,,

,, (11-34)

Also, solving Equation 13-30 for Th,out gives

( )incoutch

cinhouth TT

CCTT ,,,, −−= (11-35)

Substituting this relation into Equation 11-34 after adding and subtracting Tc,in gives

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

−

−−−+−

h

c

cincinh

incoutch

coutcincincinh

CC

CUA

TT

TTCCTTTT

1ln,,

,,,,,,

which simplifies to

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

h

c

cincinh

incoutc

h

c

CC

CUA

TTTT

CC 111ln

,,

,, (11-36)

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We now manipulate the definition of effectiveness to obtain

( )( ) cincinh

incoutc

incinh

incoutcc

QQ

TTTT

TTCTTC

QQ min

,,

,,

,,min

,,

max

εε =−−

→−−

==

Substituting this result into Equation 11-36 and solving for εgives the following relation for the effectiveness of a parallel-flow heat exchanger

ch

c

h

c

c

CC

CC

CC

CUA

minflow parallel

1

1exp1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

=ε (11-37)

or

max

min

max

min

flow parallel

1

1exp1

CC

CC

CUA

c

+

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−

=ε (11-38)

minCUA

is dimensionless

Define NTU= number of transfer units

( )minmin pCm

UACUANTU == (11-39)

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The larger the NTU, the larger the heat exchanger.

Define capacity ratio C as

max

min

CCC = (11-40)

It can be shown that

( )),NTU(function ,function maxminmin

CCCCUA

==ε

Relations for ε are given in Table 11-4 for different heat exchangers

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Figure 11.26 Effectiveness for heat exchangers

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We make the following observations from the effectiveness relations and charts given above:1) The value of the effectiveness ranges from 0 to 1

ε increases rapidly for NTU ≤ 1.5

→ large size cannot be justified economically

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Figure 11.27 For a specified NTU and capacity ratio C, the counter-flow heat exchanger has the highest effectiveness and the parallel-flow lowest.

2) For a given NTUand Cmin / Cmax

εPF < εcross flow < εCF

3) ε is independent of Cfor NTU < 0.34) ε minimum for C = 1ε maximum for C = 0

C = 0 when Cmax→∞(during a phase change process)

Figure 11.28 The effectiveness relation reduces to ε=εmax1-exp(-NTU) for all heat exchangers when the capacity ratio C = 0

During a phase change process all ε relations reduce to a single line (Figure 11.28)(in a boiler or condenser)

)exp(1max NTU−−== εε

(11-41)

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The size of a h.e. can be easily determined by using LMTD method if all temperatures are known.

It can also be determined from effectiveness-NTU method:

1) Find ε from maxQQ=ε

2) Find NTU from relations given in Table 13-5

Note:Relations in Table 11-5 and Table 11-4 are equivalent

NTU = f (ε) ε = f (NTU)

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11-6 Selection of Heat ExchangersUncertainty in the predicted value of U may be greater than 30% → Overdesign the heat exchanger.

The proper selection depends on several factors:

1) Heat Transfer Rate:

2) Cost:

- is required for the desired ΔT

- An off-the-shelf heat exchanger has a cost advantage over those made of order.

- Designing and manufacturing increase costs.

3) Pumping Power:

Operating cost = (Pumping power, kW) × (Hours of operation, h) × (Price of electricity, $/kWh)

Minimizing ΔP and will minimize the operating cost, but it will maximize the size of the heat exchanger and thus initial cost.

4) Size and Weight:

m

- The smaller and the lighter the heat exchanger, the better it is.

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5) Type:

a) type of fluidsb) size and weight limitationsc) presence of phase change processes

6) Materials:

-Depends on:

-Thermal stress effects should be considered for P > 70 atm and T > 550 oC.

For corrosive fluids → select corrosion resistance materials (stainless steel).

7) Other Considerations:

- Leak-tight,

- ease of servicing,

- low maintenance cost,

- safety and reliability

- Quietness (Important in heating and air conditioning)