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Part IIIToolbox 11:

HEAT EXCHANGER OPERATING POINT DETERMINATION

2

In the case of a condenser and evaporator, one of the fluids changes the phase and the isobaricspecific heat becomes infinite. This indicates that product (m cp) is then also infinite and the fluid

temperature passing the heat exchanger is constant. In these cases, effectiveness depends only on thenumber of transfer units (NTU). The micro and macro energy balances are equal.

2. Heat Exchangers Dimensionless Groups:The second important feature of the NTU method is the thermodynamic significance of the dimensionless groups appearing in the analysis.

The fluid heat capacity rate ratio is defined as follows:

max

min

W

W (11.3)

In this equation, Wminmeans the lower heat capacity of two streams (Wmin= min (Wh, Wc)). If thestream with lower heat capacity is recognized, then the other will be Wmax. One of the special cases

appears when Wh= Wc(the temperature changes of both streams are equal).

Simply, the ratio of the smaller to the larger heat capacity rate for the two fluid streams is asfollows:

10 (11.4)

and represents the dimensionless group suitable for understanding the overall fluid temperaturechanges. The condition = 0 indicates the tendency of the strong stream towards the isothermalchange, while = 1 the trend of each stream to undergo the same temperature change from the inlet

to the outlet of the exchanger (balanced streams).

The temperature profiles of counter-flow heat exchanger for 10 are presented pictorially

in Fig. 11.1.

Flow length

Temperature

Flow length

Temperature

Flow length

Temperature

Th,out

Tc,inWh> Wc

Th,in

Tc,out

Th,out

Tc,in

Th,in

Tc,out

Wh= Wc

Th,outTc,in

Wh< Wc

Th,in

Tc,out

Figure 11.1: Temperature Distribution in Counter-Flow Heat Exchanger

Similarly, thermodynamic reasoning can be associated with the second dimensionless group, the

number of heat transfer units:

minW

UANTU (11.5)

Simply, it is the ratio of the overall conductance UA and the smaller heat capacity rate (W min).

The range 0 NTU < in practice has a finite upper limit, but thermodynamically speaking, thehigher the NTU (higher overall conductance and smaller weak stream capacity), the smaller are thelocal temperature differences across the heat transfer surface area and consequently the irreversibilityis lower. This means that better heat exchanger flow arrangements must have a monotonically

increasing effectiveness with NTU.

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3. Heat Exchanger Effectiveness: The effectiveness () of any two-fluid heat exchanger isessentially a dimensionless measure of the heat quantity which is actually transferred between two

streams normalized with the maximum possible fluid enthalpy change in the system. Thishypothetical quantity of heat can be seen as the enthalpy change of the weak stream (stream with

lower heat capacity) undergoing the maximum possible temperature change (Th,in- Tc,in) without anylosses. The heat exchanger effectiveness is then simply defined as:

)TT(W

)TT(W

)TT(W

)TT(W

Q

Q

in,cin,hmin

in,cout,cc

in,cin,hmin

out,hin,hh

max

act (11.6)

and it is a unique measure of its thermal performance. Uniqueness in this context means that the sameeffectiveness, , is obtained by writing Qacteither in terms of hot fluid parameters or in terms of coldfluid parameters. Effectiveness is to be obtained from the solution of the mathematical modelmentioned above and will, thus, depend on two dimensionless groups which are the heat exchangerparameters NTU and .

A summary of the necessary steps to derive an effectiveness relationship is as follows:Write the differential equations describing the local heat transfer in the heat exchanger coreand specify the inlet (boundary) conditions.Identify (between the hot and cold fluid stream) the weak stream, and assign a subscript minto all its entities, assign maxto the corresponding entities of the other fluid.Solve the mathematical model by some appropriate method of calculus to obtain thetemperature distributions within the heat exchanger core.Heat exchanger effectiveness calculation.

4. Heat Exchanger Operating Point:The operating point of an exchanger is the set of , NTU and values that satisfy identically both its macroand microenergy balance. Theflow arrangementasan argument of the NTU relation makes the heat exchanger operating point unique for the

particular flow arrangement. Here, uniqueness implies that the three values (, NTU, and ) are theordered ones for the specified flow arrangement. Different flow arrangements have different operating

points even for the same values of two chosen arbitrarily out of three corresponding parameters (,NTU, and ). If this is not the case, the flow arrangements are said to be equivalent.

In practice, a designer is faced with the problem of seven physical entities (for a specific flow

arrangement and for 10 ) that have to satisfy just two equations, namely Equations (11.1) and

(11.2). These equations state an unambiguous relationship of the type:

.

cp

.

hpout,cin,cout,hin,h )tarrangemenflow,)cm(,)cm(,UA,T,T,T,T(f 0 (11.7)

For an arbitrary, but specifiedflow arrangement, any five of the seven variables must be knownfor heat exchanger operating point determination. Depending on the combination of the twounknowns that have to be determined in order to satisfy Equations (11.1) and (11.2), there are 21

possible problems for the determination of the heat exchanger operating point. They are shown inTable 11.1 classified in six groups.

It can be stated that data on mass flow rates and fluid types are included inipi

cmW (i = h, c)

or so cold strongnessof fluid streams. The units of these heat capacities are the same as for UA

[W/oC]. The dimensionless heat exchanger groups: NTU and , are combinations of these dimension

values. As overall heat transfer coefficient (U) can be defined independently of the size of heat

transfer surface area (A), complex UA has not to be divided into constituents. However, complexesWihave different nature. Known Wiassumes that both mass flow rate (mi) andisobaric specific heatof fluid (cp,i) are known. If one of these two values is not known, this means that the heat capacity is

not known.

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By defining all seven basic heat exchanger parameters according to the energy balance it ispossible to define the Heat Exchanger Operating Point (HEOP).

The sizing problems in groups I and II, and the rating problems in groups III and IV canreadily be recognized. However, the problems in groups V and VI may be described as regime

problems. They are the most difficult to solve because there is no possibility for the identification offluid streams according to their relative strongness and Equations (11.1) and (11.2) must be treatedand resolved simultaneously based upon a guess made for the Wminstream. Also, problems 14 to 21

always have one trivial (= 0) solution.

Table 11.1: Twenty-One Problems to Determine the Heat Exchanger Operating PointGroup Problem UA mhc h mcc c Th in Th out Tc in Tc out

No. No. [W/K] [W/K] [W/K] [K] [K] [K] [K]

I

1 ??? ???

2 ??? ???

3 ??? ???

4 ??? ???

II

5 ??? ???

6 ??? ???

III 7 ??? ???

IV

8 ??? ???

9 ??? ???

10 ??? ???

11 ??? ???

12 ??? ???

13 ??? ???

V

14 ??? ???

15 ??? ???

16 ??? ???

17 ??? ???

VI

18 ??? ???

19 ??? ???

20 ??? ???

21 ??? ???

In the case when = 0, there are two special types of heat exchangers named CondensersandEvaporators.

A condenser is a device in which the hot stream is converted from vapor to liquid by using the

cold stream. The hot stream heat capacity rate is then Wh and the temperature of the vapor-liquid

mixture is constant and equal to Tcon = Th,in = Th,out (isothermal change). As the cold stream heat

capacity rate is greater than zero and less than infinity (0 < Wc< ) and as the temperature of the inletcold fluid is less than outlet temperature (Tc,in< Tc,out), the heat capacity rate ratio is:

0h

c

WW (11.8)

The effectiveness of any flow arrangement of evaporators and condensers is as follows:

)NTUexp(1 (11.9)

The list of possible problems that can appear is given in Table 11.2.

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Table 11.2: Five Problems to Determine the Condensers Operating PointGroup Problem UA mccpc Tcon Tc,in Tc,out

No. No. [W/K] [W/K] [K] [K] [K]

I & II 1 ???

III & IV

2 ???

3 ???

4 ???

5 ???

An evaporator is a device in which the cold stream is converted from liquid to vapor by using the

hot stream. The cold stream heat capacity rate is then W c and the temperature of vapor-liquid

mixture is constant an equal to Teva= Tc,in= Tc,out (isothermal change). The hot stream heat capacity

rate is greater than zero and less than infinity (0 < Wh< ) and thus the heat capacity rate ratio is =0. The list of possible problems that can appear is given in Table 11.3.

Table 11.3: Five Problems to Determine the Evaporators Operating PointGroup Problem UA mhc h Th in Th out Teva

No. No. [W/K] [W/K] [K] [K] [K]

I &I I 1 ???

III & IV

2 ???

3 ???

4 ???

5 ???

5. In design work, the mean overall coefficient of heat transfer U is generally specified by the use of

design correlations for the mean coefficients of heat transfer for both sides of the heat exchanger (h h[W/(m2 oC)] and hc [W/(m

2 oC)]) in terms of Reynolds, Prandtl, etc. numbers. The nature of these

correlations is dependent on geometry, flow arrangement, and laminar or turbulent flow.

In the thermal analysis of existing heat exchangers, U can be readily determined on the basis ofrecorded measurements for terminal temperatures and flow rates by the use of solutions for heatexchanger effectiveness (). For example, can be calculated from Equation (11.7) and can beobtained from the defining relation (Eq. (8)). Known and and NTU curve, whichcorresponds to the geometry and flow arrangement of selected heat exchanger, can be used tocalculate the number of transfer units NTU. Now U can be calculated by using Equation (11.5) in theform:

NTUA

WU min (11.10)

Calculations for U obtained in either of these ways account for all of the various complicating

factors that may exist, such as fouling, fins and baffles.

6. When energy was cheap, little attention was paid to designing heat exchangers which made theoptimum use of available energy resources. Then, the primary constraints concerned the heatexchanger duty (the amount of required heat exchange and the required heat transfer surface area to

satisfy this duty). The dutyis a thermal constraint, while the sizeis an economic constraint since thecosts of materials and manufacturing are directly related to the heat transfer area.

Potential energy savings, along with economic incentives, have led to increased efforts to produceheat exchanger configurations that can be used to:

reduce the size of a heat exchanger for a specified heat duty;upgrade the capacity of an existing heat exchanger;reduce the approach temperature difference for process streams;

reduce pressure drop and pumping power.

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7. Software for Heat Exchangers Operating Point Determinationcan be used for solving any of

the 21 tasks ( 10 ) given in Table 11.1 and any of the five tasks for solving condensers (Table

11.2) and evaporators (Table 11.3). It is possible to choose one of the following flow arrangements:Parallel Flow

Counter FlowShell & Tube: One shell pass, 2, 4, 6, tube passesShell & Tube: Two shell passes, 4, 8, 12, tube passes

Cross Flow (both fluid unmixed in pass)Cross Flow (both fluid mixed in pass)Cross Flow (hot fluid (wh) mixed and cold fluid (wc) unmixed in pass)Cross Flow (Hot fluid (Wh) unmixed and Cold fluid (Wc) mixed in pass

The schemes and formulae for effectiveness calculation for all of these flow arrangements are

given in Table 11.4.The heat flow rate is defined as follows:

TUA)TT(W)TT(WQin,cout,ccout,hin,hh

(11.11)

The biggest temperature difference between streams (T) of all flow arrangements appears in thecounter flow heat exchanger (for the same inlet and outlet conditions) and it is called the logarithmicmean temperature difference. For this flow arrangement it is:

in,cout,h

out,cin,h

in,cout,hout,cin,h

ln

TT

TTln

)TT()TT(T

(11.12)

For Wh= Wcor = 1 it is:

)TT()TT(T in,cout,hout,cin,hln (11.13)

For condensers (Tc= Th,in= Th,outand hW ) it is:

in,ccon

out,ccon

in,cconout,ccon

ln

TT

TTln

)TT()TT(T

(11.14)

For evaporators (Teva= Tc,in= Tc,outand cW ) it is:

evaout,h

evain,h

evaout,hevain,h

ln

TT

TTln

)TT()TT(T

(11.15)

Software calculates this temperature difference which can be compared with the real temperature

difference for the desired flow arrangement which is also calculated by software. The ratio betweenreal and counter-flow temperatures is always less than one and gives the designer an opportunity toestimate the qualityof the heat transfer of the desired heat exchanger flow arrangement as comparedto the best one (counter-flow).

The mean temperatures of both fluids are also calculated by software. These temperatures can be

used for the calculation of mean isobaric specific fluid heat and mass flow rates. The mean pressuresof fluids are assumed to be known.

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For condensers and evaporators, the mean temperature differences are defined by Equations(11.14) and (11.15), respectively, and are equal for any flow arrangement.

The first FORM that appears in the software is used for the selection of the heat exchanger. Theoption buttons offer three possibilities: 1. General (both fluids change temperatures passing the heat

exchanger), 2. Condenser (only cold fluid changes temperature passing the heat exchanger and hotfluid is at a constant temperature changing the phase, and 3. Evaporator (only hot fluid changestemperature passing the heat exchanger and cold fluid is at constant temperature changing the phase).

After selecting the General type of heat exchanger and pressing the NEXTbutton the windowSELECTION OF FLOW ARRANGEMENT will be opened. Eight types of heat exchangerarrangements (listed above) are offered. Using the Horizontal Scroll Bar the flow arrangement canbe selected. By pressing the button NEXT, the window PROBLEM DEFINITION is opened. Byentering five known values, any of 21 problems can be solved. By using the Horizontal Scroll Bar,the problem can be selected. Text boxeswith unknown values will contain the text UNKNOWN.

After data input by clicking on the button Calculation,the program will calculate the unknown

data and display all the data including the performance of heat exchanger and heat flow rates.This program assumes that fluid stream heat capacity rates are known (in some of the problems).

This means that specific heats and mass flow rates of fluids have to be known. It is not alwayspossible to know these values. This means that they have to be assumed, and after performing thecalculation in the next window, the correction has to be made and the calculation has to be repeated.

The final window is used for the calculation of mass and volume flow rates for given fluids. Theprogram offers nine fluids to be selected and, if none corresponds to the actual fluid, the user can

input its own density and specific heat. The fluids offered by the program are:1. Hydrogen2. Nitrogen-clear3. Oxygen4. Carbon Monoxide

5. Carbon Dioxide6. Sulfur Dioxide7. Dry Air8. R717 (Ammonia)Liquid9. R22Liquid

After selecting the fluids by using the Combo Boxesand inputting the mean pressure of fluids byclicking on the button Calculation,the results appear in the proper Text Boxes.

For any correction of the calculation performed, the button Previoushas to be used.For printing the results, after finishing the calculation, the button Printing has to be used.A very similar procedure is used for the calculation of either the Condenser or Evaporator. Of

course, other fluids are offered by the program and the enthalpy of saturated vapor and boiling fluidare calculated.

Table 11.4: Effectiveness Relations for Various Heat ExchangersName and Scheme

)tarrangemenflow,,NTU(

= 0 0 < < 1 = 1

1

Single-pass parallel flow

Wh, th,in

Wh, th,out

Wc, tc,in

Wc, tc,out

)

NTU

exp(

1

1

11 )(NTUexp 2

21 NTUexp

2

Single-pass counter-flow

Wh, th,in

Wh, th,out

Wc, tc,in

Wc, tc,out

)(NTUexp

)(NTUexp

11

11

NTU1

NTU

3

Shell & Tube: one shell pass,

2, 4, 6, tube passes

Wh, th,in

Wh, th,out

Wc, tc,in

Wc, tc,out

2

2

2

11

11

11

2

NTUexp

NTUexp

21

2122

2

NTUexp

NTUexp

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4

Shell & Tube: two shell passes,

4, 8, 12, tube passes

Wh, th,in

Wh, th,out

Wc, tc,in

Wc, tc,out

2

2

1

1

11

1

where * is computed for the scheme 3 for andNTU/2

2

11

1

5

Cross-flow: both fluids unmixed

Wh, th,in Wh, th,out

Wc, tc,out

Wc, tc,in

2

210 2

122

11

n

n/n NTUINTUINTUI

NTU)(exp

NTU2INTU2I

NTU2exp1

10

6

Cross-flow: both fluids mixed


Wc, tc,out

Wc, tc,in

111 )NTUexp(

NTU

)NTUexp(

NTU

NTU

1)NTUexp(1

NTU

NTU

7

Cross-flow: Wcmixed, Wh

unmixed


Wc, tc,out

Wc, tc,in

)WW()NTUexp(exp

maxc

11

)WW()NTUexp(

exp maxh1

1 )NTUexp(exp 11

8

Cross-flow: Wcunmixed, Wh

mixed


Wc, tc,out

Wc, tc,in

)WW()NTUexp(

exp maxh1

1

)WW()NTUexp(exp

maxc

11 )NTUexp(exp 11

8. Example:A cross flow heat exchanger (both fluids unmixed) is designed to heat 2.5 kg/s of dry airat 1 bar from 15

oC to 30

oC. Hot water at 52.5

oC is used for this purpose. The outlet water

temperature is 24oC (Fig. 11.2). The mean overall coefficient of the heat transfer surface is 300

W/(m2oC)]. The other performance indicators of heat exchanger have to be determined.

Water,

Th,in= 52.5

o

C

Air,

Mc= 2.5 kg/s

pc= 1 bar

Tc,in= 15.0oC

Air,

Th,out= 30.0oC

Water,

Th,out= 24.0

o

C

Figure 11.2: Cross Flow Heat Exchanger (both fluids unmixed)

The software accompanying this Toolbox can be used for problem solving. Firstly, we select the

GENERAL option (Option Button) and by clicking on the button NEXT the second window isopened offering eight types of heat exchanger arrangement. We select flow arrangement No. 5 (Cross

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Flow Heat Exchanger: both fluids unmixed) and after that, the software switches automatically to theTable containing possible problems. We select problem No. 5. However, in our case only the mass

flow rate of air is given and we have to find the heat capacity of this stream. In the case of dry air, thespecific heat is approximately 1 kJ/(kg

oC) which means that the heat capacity is around that value. It

will be the first iteration. In the next step, the program calculates the mean fluid temperatures andspecific heat of fluids. Multiplication of this specific heat and given air mass flow rate gives the newand different heat capacity of the cold stream. By selecting the previous window and typing in the

new air heat capacity the calculation has to be repeated. In Fig. 11.3, the input data and calculatedresults are presented. It must be noticed that at this stage of the calculation the type of fluids does nothave to be known. This simplifies the problem solving as the definition of the weak and strong streamis not necessary.

Figure 11.3: PROBLEM No. 5Step 3

After the input of all the data and by clicking on the button CALCULATION, the results willappear in the frame Output Data. Now, we know the fluid heat capacities, four temperatures and thesize of the heat exchanger represented by UA. Effectiveness, NTU and are 0.7600, 2.32 and 0.5263,respectively.

The heat transfer flow rate is 37.7 kW and as we know the mean overall coefficient of the heat

transfer, we can calculate the required surface in the following way (Equation (11.9)):

]m[2.103.12300

10007.37

TU

QA 2

f

(11.16)

It is always necessary to know how the value U = 300 [W/(m2oC)] is specified (on finned or un-

finned surface, hot or cold fluid side, etc.).

The mean temperature difference between the fluids is in this case 12.3oC and it is less than the

logarithmic mean temperature difference (LMTD) which is 14.7oC. The reason for that is the use of

the cross flow heat exchanger although the best solution is a counter flow heat exchanger.The next step (step 4) in the calculation is the definition of the fluids and the determination of

fluid properties. Actually, we have already assumed the specific heat of the cold fluid and performedthe calculation in step 3. The mass flow rate of the cold stream is given and with a couple of iterationswe have to adjust the cold stream heat capacity to fit the given flow rate. For this, we have to know

the pressure of the air (cold fluid). By selecting hot fluid (Water) and its pressure (1 bar), the program

calculates the hot fluid flow rate. As cold fluid is air at the pressure of 1 bar, the program calculatesthe appropriate flow rate. The density and specific heat are calculated for the mean cold fluid

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