A novel way to design shell-and-tube heat

exchangers based on the cell method

V.G.Ramanathan Ketan Anand Nayak

April 16, 2009

1 Abstract

In this work, we present a novel technique to design shell and tube heat ex-changers based on the cell method. The conventional cell method proposed byGaddis makes the assumption that the tube side fluid is mixed. Here, we intro-duce a technique which preserves the unmixed condition of the tube side fluidduring the analysis. This technique defines a cell to comprise of only one tuberow. Since the tubes within each row suffer equal temperature gradients, thistechnique gives the exact outlet temperatures at the end of the tube. Ratherthan a single mixed tube outlet temperature, we are now able to calculate thetemperature profile at the outlet and provide insight into localized heat transferoccurring in the heat exchanger.

Keywords: baffle, cell method, tube bundle, effectiveness, multipass, cross-flow

2 Introduction

Shell-and-tube heat exchangers are the most common type of heat exchangersand are used in the chemical industry, power production, food industry, en-vironment engineering, waste heat recovery, air-conditioning and refrigeration.They are not only simple to manufacture but also versatile, robust and reliable.Due to the important role of shell-and-tube-heat exchangers, several methodshave been developed for their design. Of particular relevance to this work is thecell method of design proposed by Gaddis. This method can be used to designa single-phase multipass shell-and-tube heat exchanger with segmental baffles.However, this method only yields the adiabatic mixing temperature at the tubeoutlet and does not recover the temperature profile across the shell diameter.In this work, we suggest an improved cell method that recovers the temperatureprofile at the tube outlet.

This report is organized as follows. Section 3 gives an overview of the con-ventional cell method proposed by Gaddis. In section 4, we introduce the im-

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proved cell method and sketch an outline of the procedure. Section 5 gives thestep-by-step algorithm of the improved cell method.

3 Cell Method

Figure 1: A heat exchanger divided into a number of cells

The cell method of analysis of a shell-and-tube heat exchanger was proposedby Gaddis in 1978. This method divides the heat exchanger into multiple inter-connected cross-flow heat exchangers [4]. Figure 3 shows a shell-and-tube heatexchanger with a single tube pass and single shell pass.

The influence of the baffles on the exchanger effectiveness and on the meantemperature difference is ignored in most present thermal design calculations.This can be justified when

1. The number of baffles is large,

2. Heat capacity rates C1, and C2, (C = Mc) of the two streams differ greatlyfrom one another, or

3. Number of Transfer Units (NTU) is small.

If none of the above conditions are fulfilled, ignoring the baffle-induced shell-side flow gives a large error in computing the effectiveness of the heat exchanger.In such cases, it is possible to use numerical methods to evaluate the thermalperformance of the apparatus. The heat exchanger can be divided into a numberof cells (sub-exchangers) as shown in Fig. 1.

3.1 Assumptions in the Cell Method

• The shell and tube heat exchanger can be described using a combinationof connected heat exchangers in pure cross flow i. e, the flow in the shellside is at 90◦ to that in the tube side

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• The leakage in the shell side flow through clearances between shell andbaffle and tube and baffle are neglected.

• The flow is assumed to be uniformly distributed in the direction transverseto the flow, both for the shell side fluid and tube side fluid.

• The fluid at the exit of each cell on both the sides is assumed to becompletely mixed. This implies that the input temperature will be uniformin nature.

3.2 Division of the heat exchanger

The cell method requires the division of the shell and tube heat exchanger intoa number of connected cross flow heat exchangers. For every tube pass, thespace between adjacent baffles is considered as a cell. This kind of a divisionis primarily done because the flow is locally cross-flow due to obstruction frombaffles. This kind of a division neglects any flow accounting for baffle leakage,an assumption that has been made for this method. So the total number of cellsin a heat exchanger is given by

Number of Cells = (Number of tube passes) × (Number of Baffles) +1

So, for a shell and tube heat exchanger involving one pass on either side thenumber of cells is given by Number of Baffles +1.

Figure 2: Cross sectional view of the arrangement of tubes

3.3 Calculation procedure for conventional cell method

In the cell method the solution of one cell will be the input to another cell. Sothe calculation procedure requires the cell to be treated as a separate entity with

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a set of inputs and a set of outputs. The set of inputs may be the inlet condi-tions for the shell and tube heat exchanger or outlet condition of some other cell.

The cross flow of the shell-side fluid across the tube bank between two adja-cent baffles or between a baffle and the adjacent tube sheet is regarded as onepass. This usage should be distinguished from the more general one shell-sidepass, which means that the fluid flows from one end of the shell to the otherwith baffles treated as only a minor interruption of the basically one dimen-sional flow. Figure shows one of the heat exchanger cells considered separately.For calculation purposes, it is assumed in further calculations that stream 1 isthe stream with the lower heat capacity rate. The following equations may bewritten in the steady state [1]

Qc = C1(T′′

1 − T ′

1)

Qc = −C2(T′′

2 − T ′

2)

Qc = EcC1(T′

2 − T ′

1)

where Ec is the the temperature effectiveness of the cell, given by,

Ec =T ′′

1 − T ′

1

T ′

2 − T ′

1

Now we define non dimensional temperatures given by

Θ =T − T1i

T2i − T1i

where T1i and T2i are the heat exchanger inlet temperatures of stream 1 and2 respectively. The non dimensional temperature varies between 0 and 1. andthe dimensionless inlet temperatures are Θ1i = 0 and Θ2i = 1. Converting theabove equations to the non dimensional form, we get the following equations

Θ′′

1 = aΘ′

1 + bΘ′

2

Θ′′

2 = eΘ′

2 + fΘ′

1

where

a = 1 − Ec

b = Ec

e = 1 − REc

R =C1

C2

(0 ≤ R ≤ 1)

The numerical value of the cell effectiveness Ec, will in a general case bedifferent for each cell. It is assumed at this stage that the cell effectiveness isknown. The constants a, b, e&f can be evaluated as the value of R is known.

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These equations are applied to the Nc cells, then we obtain 2Nc equations,with 2Nc unknowns. These unknowns comprise of the output values of thetemperature of both the streams for each cell.

The inlet cell temperatures for each stream are the exit temperatures of thepreceding cells, and the dimensionless inlet temperatures of the heat exchangerare known. The solution of the system of algebraic linear equations gives therequired temperature distribution. The effectiveness E of the heat exchanger isby definition,

E =T1o − T1i

T2i − T1i= Θ1o (1)

where T1o is the outlet temperature of stream 1. The effectiveness of theheat exchanger is thus identical with the dimensionless heat exchanger outlettemperature of the stream with the lower heat capacity rate.

Figure 3: Flow arrangement in a single cell

3.4 Drawbacks

The conventional cell method as described in the above sections divides the heatexchanger into a number of connected cross-flow sub-exchangers each of whichis solved to obtain the intermediate temperatures (the output temperatures ofthe cells). As shown above, each cell is considered to have two inlets (i.e. twoinput temperatures) and two outlets (the output temperatures).

Physically each cell represents a localized region of the heat exchanger, inthis case the localized region corresponds to the region between adjacent bafflesor between a baffle and a tube end sheet . In a real scenario, considering a shelland tube heat exchanger, this region would comprise of a number of tubes ineach cell in the tube side. The conventional cell method assumes a mixed-mixedcondition for the fluids on both sides. This assumption may hold true, for acertain extent, to the shell side fluid (even though, real shell and tube heatexchangers do not have a completely mixed condition and gradients transverseto the flow are present in the shell side fluid), this assumption is totally invalidfor the tube side fluid as physical barriers make it impossible for the tube sidefluid to achieve a well mixed condition(The fluid moves in a number of tubes

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and the tubes themselves prevent the mixing of the fluids in the tube side).Also making the fluid flow unmixed in the tube side will give insight into thegradients present in the tube side outlet temperature in each cell prompting forpossibly, a better design.

So the conventional cell method previously proposed just gives a very crudemethod of analyzing heat exchangers. This is because the consideration of mix-ing condition for the tube side. Considering gradients within each cell(tube sideunmixed condition) will provide a more accurate analysis of the heat exchanger.So to overcome these drawbacks, in the following sections we propose a new,novel way of analysis, which preserves the unmixed condition of the flow on thetube side.

4 Improved Cell Method

In this section we propose a new method which helps us to calculate the tem-peratures for a unmixed-mixed scenario. This method is basically an extensionof the regular cell method to include the temperature gradient in the tube sidefluid. In this method we define the cell differently. In the new system, a cell isdefined to be the region between adjacent baffles or a baffle and a tube sheet,and containing exactly one tube row. This is contrasting to the earlier defini-tion where in we considered the entire tube bundle to be a part of the cell. Thereason why we have chosen the cell to contain only one tube row is that for thegiven cross flow condition, every tube corresponding to the same tube row willhave the same outlet temperature. As a result of this we can now formulatethe system into 2-input/2-output cells, while still obtaining a solution for theunmixed-mixed condition (temperature gradient in the tube side fluid flow)

Figure 4: Cell division in the improved cell method

Considering tube rows as cells instead of tubes will considerably simplify thesituation from the case, where individual tubes are present in a cell (one tubeper cell). To solve the given system, we need to develop an algorithm to solvefor each of the cells in a similar way as done in the conventional method.

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5 Algorithm

5.1 Solution Algorithm for the Cell

The following section presents a calculation procedure to calculate the outputtemperatures from the given input temperatures. Initially we will know both theinput temperatures only for one cell (i.e., the cell from where the shell side fluidenters or exits the heat exchanger). Now an outline for the solution procedureis provided assuming that the input temperatures to the cell are known and theoutput temperatures are unknown.

1. Make suitable assumptions for baffle spacing LB , shell diameter DS , tubeouter diameter do, tube inner diameter di and the tube pitch in the stag-gered arrangement, PT .

2. Calculate the heat transfer coefficient on the tube side for the shell usingthe appropriate correlations. For obtaining the heat transfer coefficient weneed to know the Reynolds number for the flow. The Reynolds number isgiven by,

ReD =ρ1V1di

µ1, (2)

where ρ1 is the density of the fluid in the tube side and µ1 is the viscosityof the tube side fluid. The velocity V1 is given by

V1 =m1

ρ1A1,csNtot(3)

assuming that the mass flow rate in the tube-side, m1 is equally dividedamong all the tubes. The area A1 denotes the cross-sectional area of thetube and is given by

A1,cs =πd2

i

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Ntot, is the total number of tubes in the shell and is determined by usinggeometry as the values of do, PT and DS are already assumed and henceknown.

Based on the Reynolds number obtained, the corresponding Nusselt num-ber correlation is used (laminar flow or turbulent flow). For laminar flowinside tubes Nusselt number is given by

NuD = 3.66. (4)

If the value of the Reynolds number obtained is greater than 2100, theflow is classified as a turbulent flow, and Gnielinski correlation is used toobtain the Nusselt number, which is given by [2],

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NuD =(f/8)(ReD − 1000)Pr

1 + 12.7(f/8)1/2(Pr2/3 − 1)(5)

where f is the friction factor and is obtained using

f = (0.790lnReD − 1.64)−2 3000 ≤ ReD ≤ 5 × 106 (6)

Pr is the Prandtl Number obtained from property charts. The tube sideheat transfer coefficient is obtained as

h1 =diNuD

K1(7)

K1 is the value of the conductivity for the fluid.

3. Heat Transfer Coefficient on the shell side is found out using correlationfor a bank of tubes for a staggered arrangement. To calculate this weneed to calculate the maximum velocity in the tube rows (i.e. in eachcell). This id one by considering the minimum area through which flowtakes place. This area is given by the area of the plane containing thetube centers. This area is given by,

A2 = Lc × LB − Nrow × doLB (8)

Lc is the length of the chord passing through the centers of the tubesand Nrow is the number of tubes present in the particular cell Obtainmaximum velocity Vmax as

Vmax =m2

ρA2(9)

Now the Reynolds number is calculated as

ReD =ρ2Vmaxdo

µ2(10)

where ρ2, µ2 are the corresponding properties for shell side fluid.

So Nusselt number is now obtained using the Zhukzuskas corelation

NuD = C1C2(RemD,max)(Pr0.36)

(

Pr

Prs

)1/4

(11)

Valid when,NL ≥ 20

0.7 < Pr < 500

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1000 < ReD,max < 2 × 106

C1 and m are the constants obtained from the tables. C2 is the correctionfactor for considering only one tube row and its value for a staggeredarrangement is 0.64.

The heat transfer areas in both the shell side and the tube side are calcu-lated. For the tube side the area is given by

A1 = Nrow × πdiLB (12)

On the shell side we are considering the arrangement to be similar to abank of tubes, hence the area will be given by

A2 = Nrow × πdoLB (13)

4. The overall heat transfer coefficient for the cell in cross flow is found outusing

1

UA=

1

h1A1+

1

h2A2+

t

kAm(14)

where t is the thickness of the tube, k is the thermal conductivity of thematerial of the heat exchanger. Amand t are given by

Am =A2 − A1

ln(

A2

A1

) (15)

t =do − di

2(16)

5. Find the NTU on the tube side for the cell. The NTU is given by

NTU1 =UA

(mcCp)1(17)

We have to note that while calculating the value of (mcCp)1, for the tubeside the mass flow rate is a fraction of the total tube side mass flow rate.The mass flow rate through the cell is given by

mc = m1 ×Nrow

Ntot(18)

6. Find the temperature effectiveness of the cell using the following equations[3]

K = 1 − exp(−NTU1) (19)

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R1 =mcCp,1

m2Cp,2(20)

P1 =1 − exp(−KR1)

R1(21)

We also have P1 equal to

P1 =T1,o − T1,i

T2,i − T1,i(22)

Among these now T1,o, the outlet temperature for the tube side fluid, isthe only unknown and can be found out . The second output temperatureis found by energy balance for the cell

mcCp,1(T1,o − T1,i) = m2Cp,2(T2,i − T2,o) (23)

7. The outlet temperatures of the first cell being known, is now fed into thesecond cell and the entire solution procedure is repeated, for the successivecells, till we get the final outlet temperatures of the heat exchanger.

5.2 Pressure Drop

The design of a heat exchanger in most cases involves a constraint on the pres-sure drop experienced by one or both the fluids inside the heat exchanger. Hence,once the design is completed using the solution procedure outlined above, it isnecessary to analyse whether the design conforms to the pressure drop require-ment. In case the pressure drop requirements are not being met, then the designhas to modified (by varying the physical parameters of the heat exchanger) sothat the design constraints are satisfied.

5.2.1 Shell side pressure drop

The shell side pressure drop is given by [1]

∆PS = 8jf

(

DS

de

) (

L

LB

)

ρu2s

2

(

µ

µw

)0.14

(24)

where, L is the length of the heat exchanger(here we approximate it to bethe length of the tube), deis the shell side hydraulic diameter, us is the linearvelocity, µ and µw are the viscosity of the shell side fluid at the fluid bulktemperature and the wall temperature respectively. jf is the Colburn j-factor.

The length of the heat exchanger is obtained from the product of the bafflespacing and the number of baffles. For a triangular pitch arrangement of tubesthe shell side hydraulic diameter is given by

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de =4(0.86P 2

T − 0.25πD2o)

πDo(25)

The linear velocity us is given by,

us =Gs

ρ

Gs =ms

As

As =(PT − do)DsLB

Pt

The colburn j-factor has to be obtained from tables, however, the plot islinear in log-log scale and an approximate linear equation for the plot can befound and be used for the calculation of pressure drop. The equations for thecolburn j factor are

log10 jf = 1.4 − log10 Re Re ≤ 250 (26)

log10 jf = −0.166 log10 Re − 0.6 Re > 250 (27)

5.2.2 Tube side pressure drop

The tube side pressure drop is given by [1]

∆Pt = NP

[

8jf

(

L

di

) (

µ

µw

)

−m

+ 2.5

]

ρu2t

2(28)

where NP is the number of tube side passes, ut is the velocity of the fluidin the tube. The value of m is 0.25 for Re greater than 2100 and 0.14 for Regreater than 100. An additional term of 2.5 is added in the above equationto approximately accomodate for the unaccounted effects such as contractionat inlets and expansion at outlets. The colburn j-factor is obtained from thecharts. However it is noted that the graph of the Colburn J factor is linearin the logarithmic scale and the equation for the lines can be found. So theequations for the j-factor are

log10 jf = −1.025 log10 Re − 0.925 Re ≤ 900 (29)

log10 jf = −0.23 log10 Re − 1.423 Re > 900 (30)

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5.3 Design Methodology

The design of a shell and tube heat exchangers depends on a number of param-eters like shell diameter, baffle spacing, tube diameter etc. The design require-ment may ask us to come up with a design with specific constraints. In such acase , it is difficult to obtain a valid design at the first step of the calculationitself as a lot of parameters have to be assumed. So an iterative method hasto adopted , where in the heat exchanger parameters are varied within eachiteration to satisfy all the constraints.

The initial design is done with a relatively large diameter of shell,minimumnumber of tube passes and with the greatest baffle spacing. With this condition,the criteria for the length constraint is evaluated. Then at the next stage,pressure drop conditions are checked, the initial design may satisfy just oneconstraint, or even may fail to satisfy any of the design requirement. In sucha case the variation, as suggested by a flowchart is made and the design isreevaluated considering the specifications. This iteration will go on until, allthe constraints are met and an optimum design condition has been formulated.

The number of iterations will greatly depend on the skill of the person con-ducting this analysis. Assumption of values for quantities, that are far awayfrom the design requirement will require a large number of iterations ,to achievea satisfactory design. One of the methods of avoiding this issue is to use anapproximate design method, so that an approximate value for the different pa-rameters are known before hand and they can be used to evaluate the initialdesign, so that, the initial design itself, lies closer to the required design and thenumber of iterations required from there on will be much smaller.

5.3.1 Approximate sizing

The method specified above requires the number of baffles present in the heatexchanger to be known. This in turn shall need the length of the heat exchangerto be guessed. The execution time of the algorithm will greatly depend on theaccuracy of the guess. Thus, we conduct an approximate sizing of the heatexchanger based on Kern’s method to calculate an approximate value of thenumber of baffles.

The heat transfer coefficient on the shell side is given by the correlation,

hs = 0.36k

De

(

DeGS

µ

)0.55 (

Cpµ

k

)0.33 (

µ

µw

)0.14

(31)

wherede is the shell side hydraulic diameter as described by Eqn. (25) and GS isthe shell side mass velocity obtained from

GS =mT

AS

where mT is the total mass flow on the the shell side and AS is the cross-flowarea measured along the centerline of flow.

AS =DS(PT − do)LB

PT

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The tube side heat transfer coefficient is given by Nusselt number correlationas described in Eqn. (5). The overall heat transfer coefficient is then calculated

1

U(o)A(o)=

1

h(o)A(o)+

1

h(i)A(i)+

t

kA(m)(32)

The overall heat load for the heat exchanger is calculated as follows:

Q(t) = mC(p)(T(o) − T(i)) (33)

for either side fluid. This is also equal to the heat load calculated using theoverall heat transfer

Q(t) = U(o)AδT(LM) (34)

Here, Area(A) is the entire area available for heat exchange

A = πd2

4L (35)

Thus, the length L can be calculated approximately by substituting a desiredvalue of the temperature value To

For a given baffle spacing, the number of baffles can thus be calculatedand initial design can be made using that value. It is to be noted that theapproximation is based on an approximate outlet temperature for the servicefluid which is a known desirable quantity in most cases.

References

[1] Sarit K. Das. Process Heat Transfer. Narosa Publishing House, 2005.

[2] Frank P. Incropera and David P. DeWitt. Fundamentals of Heat and Mass

Transfer. Wiley-India, 2006.

[3] R. K. Shah and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design.John-Wiley and Sons.

[4] D. Brian Spalding and J. Taborek. Heat Exchanger Design Handbook. Hemi-sphere Publishing Corporation.

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