OPTIMAL DESIGN OF DOUBLE-PIPE HEAT EXCHANGERS,

COMPARISONS

Petrik Máté1, Dr. Szepesi L. Gábor

2, Prof. Dr. Jármai Károly

3

1PhD student, University of Miskolc Department of Chemical Machinery

2associate professor, University of Miskolc Department of Chemical Machinery

3professor, University of Miskolc Department of Chemical Machinery

ABSTRACT

Heat exchangers are used in industrial and household processes to recover heat

between two process fluids. This paper shows numerical investigations on heat transfer

in a double pipe heat exchanger. The working fluids are water, and the inner and outer

tube was made from carbon steel. There are several constructions which able to

transfer the requested heat, but there is only one geometry which has the lowest cost.

This cost comes from the material cost, the fabrication cost and the operation cost.

These costs depend on the material types and different geometric sizes, for example

inner pipe diameter, outer pipe diameter, length of the tube. The performance of the

heat exchanger and the pressure drop are in a close interaction with the geometry.

Optimum sizes can be calculated from the initial conditions (when one of the process

fluid inlet and outlet temperature and the flow rate is specified). The correlations to the

Nusselt number and the friction data come from experimental studies. [1] [2]

Keywords Double-pipe heat exchanger, Heat transfer, Optimization, Comparisons

LIST OF SYMBOLS

Latin letters

A Area (m2)

C Cost ($)

c Heat capacity (J·kg-1

·K-1

)

d Diameter of inner pipe (m)

D Diameter of outer pipe (m)

f Friction factor (-)

k Overall heat transfer coefficient (W·m-2

·K-1

)

L Tube length (m)

LMTD Logarithmic mean temperature difference (°C)

m mass flow (kg·s-1

)

Nu Nusselt number (-)

Pr Prandtl number (-)

Q Heat performance (W)

Re Reynolds number (-)

T Temperature (°C)

v velocity (m s-1

)

Greek letters

α Individual heat transfer coefficient (W·m-2

·K-1

)

Δp Pressure drop (Pa)

DOI: 10.26649/musci.2017.067

η Dynamic viscosity (Pa·s)

λ Thermal conductivity (W·m-1

·K-1

)

ρ Density (kg· m-3

)

Subscripts

i inner pipe

in inlet

o outer pipe

out outlet

he heat exchanger

m mean

1. INTRODUCTION

Double pipe heat exchangers are the simplest heat transfer devices. There are two

concentric pipes with different diameters: the smaller tube inside the bigger tube. One

of the medium flows inside of the smaller tube, while the other medium flows in the

annulus. With the help of the solid wall the heat can be changed between the two

fluids, without of their direct contact. The performance of the device is depending on

three factors: the heat transfer area, the heat transfer coefficient and the mean

temperature difference. Unfortunately, this type of heat exchanger has a big

disadvantage: it has limited heat transfer area compared to a shell-and-tube heat

exchanger. This type of heat exchanger can use in a narrow range. To increase this

area, the length of the tube must also be increased. The pipe length has an effect of the

material cost and pressure drop, which causes an increased operating cost. This report

shows some possibilities of the optimal design of a double-pipe heat exchanger with

Excel Solver methods.

1.1. OBJECTIVE, VARIABLES

As all optimization tasks, an objective function must be determined at first with

different conditions. The aim of this report is to find the minimum value of the total

cost, which is the sum of the material and operational costs. Two conditions are

defined: the heat from cooling media must be equal to the heat to the heating media,

and must be equal to the performance of the heat exchanger. A heat exchanger is

usable when its performance is higher than these two values.

Of course, the change of the initial parameters means a new optimization task. For

example, the operating medium flows in the annulus instead of inside the smaller tube,

or the material of the cooling media is changed. In every case the initial parameters

must be specified.

1.2. INITIAL PARAMETERS

The main goal of this report is to find the optimal sizes and operational parameters of a

double pipe heat exchanger. The qualities of the operational medium are known: these

are the mass flow mi [kg/s], the inner temperature Ti,in [°C] and the outer temperature

Ti,out [°C]. The process medium is water. The material properties depend on the

temperature. To specify these values, the mean temperature required [3]:

2

,,

,

outiini

inm

TTT

(1)

The functions of the material properties derive from interpolation.

density of water:

3

51341138265 )102103.1103244.9105921.2109868.5100146.29094.6exp(m

kgTTTTT mmmmmT (2)

specific heat of water:

Ckg

JTTTTTc mmmmmT

1000))102589.2104699.1107411.2102819.4102638.24338.1(exp(

51341038264 (3)

heat conductivity of water:

Cm

WTTTTT mmmmmT

)104454.6105534.2103224.1104392.1101743.356528.0exp(

51341038253 (4)

dynamic viscosity of water:

sPaTTTTT mmmmmT 75124937242 10))105075.3107772.2105815.9100029.2101154.3762.9(exp( (5)

After these, the necessary heat could be calculated:

outiiniii TTmcQ ,, (6)

where ci is the specific heat capacity of the inner media’s mean temperature.

Also known the material and the inlet temperature To, in [°C] of the other medium.

1.3. VARIABLES

There are geometric and operational variables, which are used to calculate the

optimum design, the geometric values must be known. There are three values: the

diameter of the inner tube d (m), the diameter of the outer tube D (m) and the length of

the tube L (m). The wall thickness of the tubes has a constant value of 2mm. There are

another two variables, which are operational parameters: the escaping temperature

To,out [°C] and the flow rate mo [kg/s] of the cooling medium.

1.4. ANALYTICAL CONSIDERATIONS

The individual heat transfer coefficients came from Sieder and Tate [4], [5]. This

coefficient is not a material property. This depends on the velocity and the material of

the media and the geometry.

d

iiii

3

18.0

PrRe023.0 (7)

The specific geometry is the diameter of the inner tube (d). The Reynolds number

consist the velocity, which can be calculated from the mass flow rate with the help of

flow cross-section.

2

4

d

mv i

i (8)

The individual heat transfer coefficient is similar to the outer side, just the specific

geometry differs, and the material properties must be calculated to the mean

temperature of the cooling medium.

2

,,

,

outoino

outm

TTT

(9)

dD

oooo

3

18.0

PrRe023.0 (10)

In this case the cooling medium flows in the annulus, so the velocity is:

44

22

dD

mv o

o (11)

At this point the heat transfer features, the geometry and the temperatures are known,

so the performance of the device can be calculated. The heat transfer coefficient

depends on the convection of the inner tube both side and the conduction in the wall.

osteel

steel

i

sk

11

1

(12)

The heat transfer area depends on the diameter and the length of the inner tube. It does

not depend on the diameter of the outer pipe, because the two process fluids contact

each other at the surface of the inner pipe. So, the bigger outer pipe does not increase

the heat transfer area. Increasing of the outer pipe diameter causes higher material cost

and lower heat convection performance.

LdA (13)

At the beginning of the calculation must be decided the flow paths. In this case that

will be clear parallel flow or clear counter flow. In case of counter flow the average

temperature-difference is smoother than in parallel case. That means a higher driving

force to the device.

Figure 2: Comparisons of parallel flow and counter flow.

outoouti

inoini

outooutiinoini

pf

TT

TT

TTTTLMDT

,,

,,

,,,,

ln

)()(

(14)

inoouti

outoini

inooutioutoini

cf

TT

TT

TTTTLMDT

,,

,,

,,,,

ln

)()(

(15)

At this point, the heat of the cooling medium and the performance of the heat

exchanger can be calculated on the following way:

inooutooocm TTmcQ ,, (16)

LMDTAkQhe (17)

The two conditions to the optimal design are the followings:

QQcond cm :1 (18)

QQcond he :2 (19)

These two conditions mean the heat which the process fluid give up must be equal

with the heat which the cooling fluid take up and must be equal with the performance

of the heat exchanger. If these two conditions are satisfied, the device is able to

transfer the necessary heat.

2. OBJECTIVE FUNCTION

2.1. MATERIAL COST

The material cost is directly proportional with the mass of the tubes. The most

common used structural materials are the carbon steel, austenitic steel, aluminum and

copper. The corrosion properties of the process fluids define the type of the material.

Over this corrosion property, these metals have different heat conductivity and

different cost. So, the material selection is part of the initial parameters. During the

optimization, the material grade is constant.

LDsDdsd

scC matMm

44

)2(

44

)2( 2222 (20)

In this correlation, cM means a specific material cost [$/kg], ρm is the density of the

structural material [kg/m3] and s is wall thickness [mm] (this is constant).

2.2. OPERATIONAL COST

Beside the material cost, another important factor is the operational cost. Flow created

frictional forces, which will restrain the flow. Due to this fact, between the two ends of

the pipe will be a pressure difference. This pressure drop (Δp) is a loss in the system. If

bigger the pressure drop, higher the operational cost and to complement this energy, a

pump must be used. The pressure drop depends on a friction factor, the velocity and

density of the stream and the geometric sizes of the pipe.

The friction factor came from the Blasius equation:

,Re

316.025.0

i

if (21)

to the inner tube and

,Re

316.025.0

o

of (22)

to the outer tube. The pressure drop in the straight line is the following:

,2

2

iiii

v

d

Lfp

(23)

.2

2

oooo

v

dD

Lfp

(24)

It is clearly seen, that the value of the pressure drop depends on the variables, just as

the conditions. That means there are geometric and operating parameters, where the

conditions are satisfied and the objective has a minimum value.

Unfortunately to build a very long heat exchanger is not possibly, so a maximum

length must be defined. The longer the length, the more the necessary elbows. This

study maximized the length at the value of 3meters. So, the numbers of elbows are

rounding down the ratio of the length and the maximized length.

roundingm

Lne

3 (25)

The pressure drop in this elbows is the function of the velocity, the density and a

coefficient, which came from Fábry: [6]

2

12.12.02

,ii

eei

vnp

and .

212.12.0

2

,oo

eeo

vnp

(26,27)

The total operation cost is the next:

).()( ,, eoo

o

oEeii

i

iEop pp

mcpp

mcC

(28)

where is cE is a specific operational cost ($/kWh). The total cost is the amount of the

material and operational costs:

opmtotal CCC . (29)

2.3. BOUNDARY CONDITIONS

The diameter of inner tube must be at least 25mm. Under this size, the inner velocity

will too big value, which occurs a fast erosion in the tube wall. Naturally, the outer

tube must be bigger diameter than the inner one. If these two diameters are close

together, the velocity in the annulus will be high, this could be also harmful.

The escaping temperature of the outer media must be a higher value than the entering

temperature. The flow rate and the tube length must be positive numbers.

3. DESIGN EXAMPLE

3.1. WATER AND WATER MEDIA, TECHNICAL FLUID IN THE INNER TUBE

A double pipe heat exchanger must be build, when the technical fluid is water, which

flows in the inner tube, the inlet flow rate is 2kg/s, the inlet temperature is 80°C and

the outlet temperature is 50°C. The cooling stream is water flows in the annulus and its

inlet temperature is 20°C. The media flow counter-current in the first case and parallel

current in the second case.

Table 1: Optimized values to a heat exchanger using water to water

Counter-current Parallel-current

Inner tube d 73.63mm 75,79mm

Outer tube D 114.28mm 132,59mm

Mass flow mo 2.75 kg/s 4,41 kg/s

Outlet

temperature To,out 41.78 °C 33,61°C

Length L 29.99m 31,92m

Material cost Cm 681.56$ 802,57$

Operational cost Cop 297.07$ 341,28$

Total cost Ctotal 978.62$ 1143,86$

3.2. WATER AND WATER MEDIA, TECHNICAL FLUID IN THE ANNULUS

There is another optimization process, when the technical fluid flows in the annulus

instead of the inner tube. That means the boundary conditions are changed, which

occur different optimum point. The optimum points are calculated counter-current and

parallel-current.

Table 2: Optimized values to a water to water heat exchanger

Counter-current Parallel-current

Inner tube d 165.58mm 179.73mm

Outer tube D 186.57mm 199.79mm

Mass flow mo 12.34 kg/s 14.43 kg/s

Outlet

temperature To,out 24.86 °C 24.16°C

Length L 11.99m 11.47m

Material cost Cm 505.93$ 520.81$

Operational cost Cop 220.36$ 224.83$

Total cost Ctotal 726.29$ 745.64$

In Table 1 and Table 2 it can be seen, that the counter-current is better than parallel

current and the correct selection of the flow area effects the optimum point.

3.3. WATER AND ETHANOL MEDIA, TECHNICAL FLUID IN THE INNER

TUBE

Uncommon in industrial practice that water heat or cool other water stream. For

example, in a distillation ethanol stream must be cooled in a heat exchanger. In this

case, the material properties are changed. The properties of the ethanol can be

calculated as the water’s properties, just the polynomial coefficients are different.

Table 3: Polynomial coefficients to the material properties of ethanol a0 a1 a2 a3 a4 a5

ρ -0.1079 -7.7201·10-3

1.5906·10-4

-1.6139·10-6

7.1873·10-9

-1.2075·10-11

c 0.81763 2.6793·10-3

1.3888·10-5

-4.3856·10-11

4.4424·10-10

1.5104·10-12

λ -1.6976 -1.2503·10-3

7.5291·10-7

5.2361·10-8

-3.4986·10-10

6.4599·10-13

η 0.5894 -2.2540·10-2

1.0283·10-4

-8.8574·10-7

4.7884·10-9

-9.7493·10-12

3

5

5

4

4

3

3

2

210)( )exp(m

kgTaTaTaTaTaa mmmmmethT (30)

Kkg

JTaTaTaTaTaac mmmmmethT

1000)exp(

5

5

4

4

3

3

2

210)( (31)

Km

WTaTaTaTaTaa mmmmmethT

)exp(

5

5

4

4

3

3

2

210)( (32)

sPaTaTaTaTaTaa mmmmmethT 35

5

4

4

3

3

2

210)( 10)exp( (33)

So, the initial conditions are the same as the 3.1. section, but the material of the fluid is

ethanol. After optimization, the results are the following:

Table 4: Optimized values to an ethanol to water heat exchanger, technical fluid in the

inner tube

Counter-current Parallel-current

Inner tube d 70.99mm 65.43mm

Outer tube D 111.24mm 121.63mm

Mass flow mo 2.25 kg/s 3.51 kg/s

Outlet

temperature To,out 38.07 °C 31.6°C

Length L 38.8m 44.99m

Material cost Cm 855.74$ 1017.78$

Operational cost Cop 371.69$ 459.59$

Total cost Ctotal 1227.73$ 1477.37$

If the ethanol stream pass into the annulus instead of the inner tube, the results:

Table 5: Optimized values to an ethanol to water heat exchanger, technical fluid in the

annulus

Counter-current Parallel-current

Inner tube d 232.15mm 247.67mm

Outer tube D 249.13mm 263.94mm

Mass flow mo 19.8 kg/s 22.35 kg/s

Outlet

temperature To,out 22.05 °C 21.82°C

Length L 10.67m 10.15m

Material cost Cm 613.35$ 619.81$

Operational cost Cop 275.12$ 277.95$

Total cost Ctotal 888.47$ 897.76$

It is true for this material-pair, that the counter-current is more favourable than the

parallel-current. From Table 1, 2, 4 and 5, the results show that we get a smaller

optimum value

, if the technical stream flows in the annulus. Calculation shows The diameter of the

tubes getting bigger, but the length of the tubes getting smaller. The fluid velocities in

every case is lesser than 1 m/s. If the corrosion is negligible, worth to calculate the

performance at the case of the two possible ways.

4. COMPARISONS, INITIAL PARAMETER SENSIBILITY

4.1. SENSIBILITY OF THE MASS FLOW RATE

Table 6: Optimum results in the function of the mass flow rate mi d D mo To.out L Cm Cop Ctotal

0.25 26.52 42.80 0.34 42.30 8.99 69.55 29.58 99.13

0.50 37.18 58.99 0.68 42.23 13.46 146.86 64.06 210.92

0.75 45.30 71.39 1.01 42.24 17.09 228.01 99.22 327.23

1.00 52.16 81.81 1.35 42.30 20.26 311.72 135.45 447.17

1.25 58.22 90.97 1.68 42.36 23.12 397.43 172.66 570.09

1.50 63.73 99.26 2.01 42.44 25.77 485.04 210.57 695.61

1.75 68.82 106.88 2.34 42.52 28.25 574.23 249.24 823.47

2.00 73.58 113.98 2.66 42.61 30.60 664.94 288.55 953.49

2.25 77.85 120.50 2.99 42.59 32.77 754.01 327.10 1081.11

2.50 82.13 126.82 3.31 42.68 34.92 847.19 367.39 1214.58

2.75 86.20 132.82 3.63 42.77 36.99 941.48 408.40 1349.87

3.00 89.87 138.41 3.96 42.76 38.91 1032.89 448.14 1481.03

3.25 93.65 143.94 4.28 42.84 40.84 1129.39 489.89 1619.27

3.50 97.31 149.28 4.59 42.92 42.72 1226.85 532.29 1759.14

3.75 100.58 154.23 4.92 42.92 44.48 1320.49 572.80 1893.29

4.00 104.02 159.22 5.23 43.00 46.27 1419.80 616.02 2035.82

4.25 107.08 163.86 5.55 43.00 47.93 1514.49 657.10 2171.59

4.50 110.35 168.58 5.86 43.08 49.64 1615.59 701.05 2316.64

4.75 113.56 173.18 6.16 43.16 51.32 1717.68 745.43 2463.11

5.00 116.38 177.45 6.49 43.16 52.88 1814.09 787.27 2601.36

Table 6 shows the optimization results in the function of the mass flow rate, in case of

water to water heat transfer. According to this table, if the inlet mass flow is

increasing, the variables are mostly increasing, however, the relation is not linear.

4.2. SENSIBILITY OF THE OUTLET TEMPERATURE

Table 7: Optimum results in the function of escaping temperature

To.in d D mo To.out L Cm Cop Ctot

70 71.63 91.90 1.01 39.89 7.07 133.57 58.12 191.69

65 72.03 98.12 1.44 40.86 11.61 228.36 99.15 327.51

60 72.71 103.90 1.85 41.65 16.92 345.74 149.99 495.73

55 73.11 109.03 2.26 42.19 23.15 488.11 211.75 699.87

50 73.63 114.29 2.76 41.79 30.00 653.15 297.07 950.22

45 73.91 118.95 3.17 42.13 39.00 871.93 392.99 1264.92

40 74.02 123.21 3.51 42.81 50.94 1165.14 505.89 1671.03

35 74.29 127.93 3.97 42.71 65.93 1547.03 671.94 2218.97

30 75.10 132.52 4.44 42.52 87.00 2097.03 926.39 3023.42

25 74.92 138.67 5.07 41.72 125.00 3101.34 1349.34 4450.68

CONCLUSION

Despite of that the double pipe heat exchanger is the simplest of the all heat exchanger,

its optimal design is difficult. In this study, there are only five variables: the diameter

of the inner and outer tube, the mass flow rate, the outlet temperature and the tube

length. As a pressure vessel, the wall thickness of tube depends on the operating

pressures, but in this case, there were constants. Use of standard sizes causes smaller

material costs, because it does not roll the tube from sheet plate, but available in the

market. Further conclusions:

The counter-current is always more favourable than the parallel current.

Technical medium in the annulus cause smaller optimum value point.

0,00

50,00

100,00

150,00

200,00

0,00 1,00 2,00 3,00 4,00 5,00

Change of mass flow rate (optimum values of d and D)

d D

0,00

500,00

1000,00

1500,00

2000,00

2500,00

3000,00

0,00 1,00 2,00 3,00 4,00 5,00

Change of mass flow rate (optimum values of costs)

Cm Cop Ctot

ACKNOWLEDGEMENT

The research was supported by the Hungarian Scientific Research Fund OTKA T

109860 project and the study was carried out as part of the EFOP-3.6.1-16-00011

“Younger and Renewing University – Innovative Knowledge City – institutional

development of the University of Miskolc aiming at intelligent specialisation” project

implemented in the framework of the Széchenyi 2020 program. The realization of this

project is supported by the European Union, co-financed by the European Social

Fund.”

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2016.

[3] M. Sheikholeslami and D. D. Ganji, “Fluid flow and heat transfer in an air-to-

water double-pipe heat exchanger,” pp. 1–12, 2015.

[4] E. N. Sieder and G. E. Tate, “Heat transfer and pressure drop of liquids in tube,”

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[5] P. K. Swamee, N. Aggarwal, and V. Aggarwal, “Optimum design of double pipe

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[6] G. Fábry, Vegipari gépészek kézikönyve. 1987.