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ME 437: Design and Rating of Heat Exchangers 1.1

CASE STUDY 1

Introduction to Heat Exchangers: Types, Classification and Analysis

Objectives

In this chapter we will introduce heat exchangers that transfer heat from a hot fluid to thecold fluid; different types of heat exchangers will be introduced; in addition to the

concept of total thermal resistance or overall heat-transfer coefficient in exchanger

analysis as well as the log-mean-temperature difference method of analysis for design

calculations.

1.1 Introduction

The technology of heating and cooling of systems is one of the most basic areas of

mechanical engineering. Wherever steam is used, or wherever hot or cold fluids arerequired we will find a heat exchanger. They are used to heat and cool homes, offices,markets, shopping malls, cars, trucks, trailers, aero-planes, and other transportation

systems. They are used to process foods, paper, petroleum, and in many other industrial

processes. They are found in superconductors, fusion power labs, spacecrafts, andadvanced computer systems. The list of applications, in both low and high tech industries,

is practically endless.

In our basic study of thermodynamics and heat transfer, we studied the form of thecontrol volume energy balance and its application too many engineering problems,including to a basic heat exchanger problem. In this module, we will extend heat

exchanger analysis to include the convection rate equation, and demonstrate themethodology for predicting heat exchanger performance that include both design and

performance rating problems.

1.2 Heat Exchanger Types

Heat exchangers are typically classified according to flow arrangement and type of

construction. In this introductory treatment, we will consider three types that arerepresentative of a wide variety of exchangers used in industrial practice. The simplest

heat exchanger is one for which the hot and cold fluids flow in the same or opposite

directions in a concentric-tube (or double-pipe) construction. In the parallel-flowarrangement of Figure 1.1a, the hot and cold fluids enter at the same end, flow in thesame direction, and leave at the same end. In the counterflow arrangement, Figure 1.1b,

the fluids enter at opposite ends, flow in opposite directions, and leave at opposite ends.

A common configuration for power plant and large industrial applications is the shell-

and-tube heat exchanger, shown in Figure 1.1c. This exchanger has one shell with

multiple tubes, but the flow makes one pass through the shell. Baffles are usually

installed to increase the convection coefficient of the shell side by inducing turbulence

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and a cross-flow velocity component. The cross-flow heat exchanger, Figure 1.1d, is

constructed with a stack of thin plates bonded to a series of parallel tubes. The platesfunction as fins to enhance convection heat transfer and to ensure cross-flow over the

tubes. Usually it is a gas that flows over the fin surfaces and the tubes, while a liquid

flows in the tube. Such exchangers are used for air-conditioner and refrigeration heat

rejection applications.

Figure 1.1 Types of heat exchangers (a) concentric-tube parallel-flow; (b) concentric-tube counter-flow; (c) shell-and-tube; and (d) cross flow.

1.3 Heat Exchanger Analysis Overall Heat Transfer Coefficient

The total heat transfer resistance is the sum of the individual components as shown in

Figure 2; i.e., resistance of the inside flow, the conduction resistance in the tube material,and the outside convective resistance, given by

ooii

totalhAkA

t

hAR

UA

111

ln

++== (1.1)

where subscripts iand orefer to inner and outer heat-transfer surface areas, respectively, t

is the wall thickness, and is the logarithmic mean heat transfer area, defined aslnA

=

i

o

io

AA

AAA

ln

)(ln (1.2)

The total heat transfer resistance can be defined in terms of overall heat transfer

coefficient based on either outer or inner areas, as long as the basis is clearly spelled out.For example, based on outer area, we have

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Figure 1.2 Typical heat transfer resistances when heat flows from a hot to cold fluid.

o

o

ii

o

totalo

o hkA

At

hA

ARA

U

11

ln

++== (1.3)

which after simplifying yields the overall heat transfer coefficient based on inner and

outer areas, respectively as

oo

iioi

i

i

hD

D

k

DDD

h

U

++

=

2

)/ln(1

1 (1.4)

and

o

ioo

ii

o

o

hk

DDD

hD

DU

1

2

)/ln(1

++

= (1.5)

where the inner and outer heat-transfer areas, as well as the wall thickness, and thelogarithmic mean heat transfer area, in terms of tube inner and out diameters and length

L, are given, respectively, as

LDA ii = (1.6.1)

LDA oo = (1.6.2)

2

io DDt

= (1.6.3)

=

i

o

io

DD

LDDA

ln

)(ln

(1.6.4)

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We note from the above equations that if the wall thickness is negligible , for

example, in thin tube heat exchangers or the thermal conductivity of the tube material is

very high, the conduction resistance through the tube may be neglected in Equations (1.4)and (1.5) to give

)( io DD

oioi hhUU

1111+= (1.7)

The convection coefficients for the inlet and outlet side of the heat exchanger tube can be

estimated using empirical correlations appropriate for the flow geometry and conditions.

During normal heat exchanger operation, surfaces are subjected to fouling by fluid

impurities, rust formation, and scale depositions, which can markedly increase theresistance to heat transfer between the fluids. For such situations, one would add the

fouling resistance (inside and/or outside-side) to Equations 1.3 to give

oo

of

i

if

ii

totalhAA

R

kA

t

A

R

hAR

11

0

"

,

ln

"

,++++= (1.8)

where and are the inside and outside fouling resistances per unit respective

heat-transfer areas, in m

"

,ifR"

,ofR2.K/W. In actual applications, fouling is normally on one-side of

the heat-transfer surface. Therefore, if the overall heat transfer coefficient based on theclean condition is, Uc, determined typically based on outer tube area (refer to Equation

1.5), the time-dependent U based on fouled condition can be written as

)(1

)(

1 '' tRUtU

f

cf

+= (1.9)

It is important to emphasize thatfouling or scalingis the most common in heat exchangerapplications. For instance, scaling is typically associated with inverse solubility salts,

such as CaCO3, CaSO4, Ca3(PO4)2, CaSiO3, Ca(OH)2, Mg(OH)2, MgSiO3, Na2SO4,

LiSO4, and Li2CO3. The characteristic which is termed inverse solubility is that, unlikemost inorganic materials, the solubility decreaseswith temperature. The most important

of these compounds is calcium carbonate, CaCO3. Calcium carbonate exists in several

forms, but one of the more important is limestone. As water runs through aquifers,

running primarily through openings in limestone rock, it becomes saturated with calciumcarbonate. This saturated water if pumped from the ground and passed through a heat

exchanger, becomes supersaturated as it is heated, so that CaCO3 begins to crystallize on

heat exchanger internal passages. Similar results occur when ground water is used in anyindustrial cooling process.

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Most of the actual data on fouling resistances or fouling factors is tightly held be a fewspecialty consulting companies. The data which is commonly available is sparse. An

example is shown in Table 1.1 for some typical industrial applications.

Note that the data in the table is given to only one significant figure. It is standard

engineering practice to indicate the precision of a number by the number of significantfigures presented, i.e. 0.0001 would indicate a number between 0.00005 and 0.00015.

Actually these numbers are not known to this precision. Nevertheless, this data is oftenthe best that is openly available and it is used for heat exchanger design calculations.

Table 1.1 Representative Fouling Factors from Incropera and DeWitt [1]

Fluid "fR ,

m2.K/WSeawater and treated boiler feed-water (below 50oC) 0.0001

Seawater and treated boiler feed-water (above 50oC) 0.0002

River water (below 50oC) 0.0002-0.001

Fuel Oil 0.0009

Refrigerating liquids 0.0002

Steam (non-oil bearing) 0.0001

1.4 Energy Equations for Heat Exchangers

The general heat exchanger equation is written in terms of the mean-temperature

difference between the hot and cold fluid, mT , as

mTUAQ = (1.10)

This equation, combined with the First Law equations, defines the energy flows for a heat

exchanger (refer to Figure 2). For the hot fluid the First Law equation is written in terms

of the temperature change the hot fluid undergoes, hT as:

hhhhp TCTCmQ == && )( (1.11)

And for the cold fluid the First Law equation can be written in terms of the cold fluid

temperature change, as:cT

ccccp TCTCmQ == && )( (1.12)

where and are the hot and cold fluid capacitance rates, respectively.hC&

cC&

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1.5 Log Mean Temperature Differences

Heat flows between the hot and cold streams due to the temperature difference across the

tube acting as a driving force. As seen in Figure 3, the difference will vary with axial

location so that one must speak in terms of the effective or integrated averagetemperature differences.

Figure 1.3 Temperature differences between hot and cold process streams

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The form of the average temperature difference, mT , may be determined by applying an

energy balance to differential control volumes (elements ) in the hot and cold fluids. As

shown in Figure 1.3, for the case of parallel flow arrangement, each element is of lengthdx and the heat transfer surface area is dA. It follows for the hot fluid

hhdTCdQ &= (1.13)

and for the cold fluid

ccdTCdQ &= (1.14)

The heat transfer across the surface area dA may be expressed by the convection rateequation in differential form

dTUdAdQ= (1.15)

where dT =Th- Tcis the local temperature difference between the hot and cold fluids.

To determine the integrated form of Equation 1.15, we begin by substituting Equations

1.13 and 1.14 into the differential form for the temperature difference,

)()( ch TTdTd = (1.16)

to obtain

+==

ch

chCC

dQdTdTTd&&

11)( (1.17)

Substituting for dQ from Equation 1.115 and integrating across the exchanger area, we

obtain

+=

A

ch

dACC

UT

Td

0

2

1

11)(

&&

+=

ch CCUA

T

T

&&

11ln

1

2 (1.18)

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)( 0,,1 cih TTT = and )( ,,2 icoh TTT = (1.22)

As discussed above, the effective mean temperature difference calculated from thisequation is known as the log mean temperature difference, frequently abbreviated as

LMTD, based on the type of mathematical average which it describes. While theequation applies to either parallel or counter flow, it can be shown that Tmwill alwaysbe greater in the counter flow arrangement. This can be shown theoretically from Second

Law considerations but, for the undergraduate student, it is generally more satisfying to

arbitrarily choose a set of temperatures and check the results from the two equations. The

only restrictions that we place on the case are that it be physically possible for parallel

flow, i.e. 1 and 2must both be positive.

Example 1.1: We are interested in cooling a light oil (this time with Cp= 1100J/kg K,

just to make it interesting) with water in a heat exchanger. The oil flows at 0.5 kg/sand is

cooled from 375Kto 350K. The water starts out at 280Kand flows at 0.2 kg/s. We would

like to determine the difference in area between a co-current (parallel flow) exchangerand a counter-current exchanger, if the overall heat transfer coefficient is 100 W/m

2K.

Solution

Known: Fluid flow rates and inlet temperatures for a heat exchanger (having either

counter-flow or parallel-flow arrangement), as well as the outlet hot flow temperature.

Schematic and Given Data:

Specific heat of light oil, Cp= 1100J/kg K; and the overall heat transfer coefficient of the

heat exchanger under both configuration, U= 100 W/m2K.

Assumptions:

Negligible heat loss to the surroundings.

Negligible kinetic and potential energy effects.

No shaft work.

Constant properties.

Negligible tube wall thermal resistance and fouling factors.

Fully developed conditions for water flow.

Analysis:

Note that, in this problem hot fluid is light oil while the cold fluid is water. We start with

a macroscopic thermal energy balance (refer to Equations (1.11) and 1.12) to find thewater outlet temperature. This gives

ic

cp

ohihhp

oc TCm

TTCmT ,

,,

,)(

)()(+

=

&

&

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Plugging the numbers, we get

K296.37K280kJ/kg.K)kg/s)(4.2(0.2

350)K(375kJ/kg.K)kg/s)(1.1(0.5, =+

=

ocT

We then need to calculate the total heat duty of the exchanger either by Equation (1.11)or (1.12), to give

cchh TCTCQ == &&

or

kW13.75K280)96.37kJ/kg.K)(2kg/s)(4.2(0.2

K375)50kJ/kg.K)(3kg/s)(1.1(0.5Q

==

=

Finally, we use the heat duty, the overall heat transfer coefficient, and the log-meantemperature difference to obtain the necessary heat-transfer areas for both the

configurations. The heat duty is given by Equation 1.21, to give

=

1

2

12

ln

)(

TT

TTUAQ

or, it can be arranged to give

lmTU

QA

=

where

Tlmis given by

=

1

2

12

ln

)(

TT

TTTlm

For counter-flow, we have

K78.63K296.37)(375)( 0,,1 === cih TTT

K70K280)(350)( ,,2 === icoh TTT

and finally the log-mean-temperature difference and heat exchanger area as

( )K74.23

78.6370ln

78.63)(70=

= lmT ,

2

2m1.85

K)K)(74.23W/m(100

W1,3750==A

For parallel-flow heat exchanger, we have

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K95K280)(375)( ,,1 === icih TTT

K53.63K296.37)(350)( ,,2 === ocoh TTT

and the log-mean-temperature difference

( )K72.35

53.6395ln

53.63)(95=

= lmT

This gives

2

2m1.90

K)K)(72.35W/m(100

W1,3750==A

This simple example shows that the counter-current system necessitates a (slightly)smaller area. Another interesting observation from the above example is that counter flow

is more appropriate for maximum energy recovery. In a number of industrial applicationsthere will be considerable energy available within a hot waste stream which may berecovered before the stream is discharged. This is done by recovering energy into a fresh

cold stream. Note in the Figures shown above that the hot stream may be cooled to Tc,i

for counter flow, but may only be cooled to Tc,ofor parallel flow. Counter flow allowsfor a greater degree of energy recovery. Similar arguments may be made to show the

advantage of counter flow for energy recovery from refrigerated cold streams.

1.4.1 Applications for Counter and Parallel Flows

We have seen two advantages for counter flow, (a) larger effective LMTD and (b) greaterpotential energy recovery. The advantage of the larger LMTD, as seen from the heat

exchanger equation, is that a larger LMTD permits a smaller heat exchanger area, A, for a

given thermal duty, Q. This would normally be expected to result in smaller, lessexpensive equipment for a given application.

This should not lead to the assumption that counter flow is always a superior. Parallelflows are advantageous (a) where the high initial heating rate may be used to advantage

and (b) where the more moderate temperatures developed at the tube walls are required.

In heating very viscous fluids, parallel flow provides for rapid initial heating. The quickdecrease in viscosity which results may significantly reduce pumping requirements

through the heat exchanger. The decrease in viscosity also serves to shorten the distance

required for flow to transition from laminar to turbulent, enhancing heat transfer rates.

Where the improvements in heat transfer rates compensate for the lower LMTD parallelflow may be used to advantage.

A second feature of parallel flow may occur due to the moderation of tube walltemperatures. As an example, consider a case where convective coefficients are

approximately equal on both sides of the heat exchanger tube. This will result in the

tube wall temperatures being about the average of the two stream temperatures. In the

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case of counter flow the two extreme hot temperatures are at one end, the two extremecold temperatures at the other. This produces relatively hot tube wall temperatures at one

end and relatively cold temperatures at the other. Temperature sensitive fluids, notably

food products, pharmaceuticals and biological products, are less likely to be scorched

or thermally damaged in a parallel flow heat exchanger. Chemical reaction fouling

may be considered as leading to a thermally damaged process stream. In such cases,counter flow may result in greater fouling rates and, ultimately, lower thermal

performance. Other types of fouling are also thermally sensitive. Most notable arescaling, corrosion fouling and freezing fouling.

1.5 Special Heat Exchanger Operating Conditions

In Figure 1.4, the temperature distributions associated with three special conditions under

which heat exchangers may be operated, are shown.

Ch> Cc. For this case, the hot fluid capacity rate Ch is much larger than the cold fluid

capacity rate Cc. As shown in Figure 1.4a, the hot fluid temperature remainsapproximately constant throughout the exchanger, while the temperature of the cold fluidincreases. The same condition could be achieved if the hot fluid is a condensing vapor.

Condensation occurs at a constant temperature, and for all practical purposes, Ch.

Ch < Cc. For this case, as shown in Figure 1.4b, the cold fluid temperature remains

approximately constant throughout the exchanger, while the temperature of the hot fluiddecreases. The same effect is achieved if the cold fluid experiences evaporation for

which Cc.Note thatwith evaporation and condensation, the fluid energy balances

can only be written in terms of the phase change enthalpies.

Ch = Cc. The third case, Figure 1.4c, involves a counter-flow exchanger for which theheat capacity rates are equal. The temperature difference T must be constantthroughout the exchanger, in which case, .21 mTTT ==

Figure 1.4 Special heat exchanger conditions (a) Ch>> Cc; (b)Ch


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1.6 References

[1] Incropera, F. P. and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 5th

Edition, John Wiley and Sons Inc., New York, 2002.

[2] Thomas, L.C.,Heat Transfer Professional Version, 2nd edition, CapstonePublishing Corp, 1999.

[3] engel, Yunus A.,Heat Transfer: A Practical Approach, 2nd edition, McGraw-Hill, New York, 2003.

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