OPTIMAL DESIGN OF DOUBLE-PIPE HEAT EXCHANGERS, COMPARISONS
Petrik Máté1, Dr. Szepesi L. Gábor2, Prof. Dr. Jármai Károly3
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
, , ,
out i in i in m
T
T T
(1)
The functions of the material properties derive from interpolation.
density of water:
3 13 5 11 4
8 3 6 2
5 5.986810 2.592110 9.324410 1.210310 )
10 0146 . 2 9094 . 6
exp( m
T kg T
T T
Tm m m m m
T
(2)
specific heat of water:
C kg T J
T T
T T
cT m m m m m
(exp(1.4338 2.2638104 4.2819106 2 2.7411108 3 1.46991010 4 2.25891013 5))1000 (3)
heat conductivity of water:
C m T W T
T T
Tm m m m m
Texp(0.565283.1743103 1.4392105 21.3224108 32.55341010 46.44541013 5)
(4)
dynamic viscosity of water:
s Pa T
T T
T
Tm m m m m
T(exp(9.7623.1154102 2.0029104 29.5815107 32.7772109 43.50751012 5))107
(5)
After these, the necessary heat could be calculated:
iin iout
i
i m T T
c
Q , , (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
i i i i
3
8 1 .
0 Pr
Re 023 .
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.
42 d
vi mi (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
, , ,
out o in o out m
T
T T
(9)
d D
o o o
o
3
8 1 .
0 Pr
Re 023 .
0 (10)
In this case the cooling medium flows in the annulus, so the velocity is:
4 4
2
2
D d
vo mo (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.
o steel steel i
k s
1 1
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.
L d
A (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.
out o out i
in o in i
out o out i in o in i pf
T T
T T
T T T LMDT T
, ,
, ,
, , , ,
ln
) (
) (
(14)
in o out i
out o in i
in o out i out o in i cf
T T
T T
T T T
LMDT T
, ,
, ,
, , ,
,
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:
oout oin
o o
cm c m T T
Q , , (16)
LMDT A
k
Qhe (17)
The two conditions to the optimal design are the followings:
Q Q
cond1: cm (18)
Q Q
cond2: he (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.
D L s D d
s s d
c
Cm M mat
4 4
) 2 ( 4 4
) 2
( 2 2 2 2
(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 . 0
25 . 0 i
fi (21)
to the inner tube and
Re , 316 . 0
25 . 0 o
fo (22)
to the outer tube. The pressure drop in the straight line is the following:
2 ,
2 i i i i
v d f L
p
(23)
2 .
2 o o o
o
v d D f L
p
(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.
rounding m
ne L 3
(25)
The pressure drop in this elbows is the function of the velocity, the density and a coefficient, which came from Fábry: [6]
12 2 . 1 2 . 0
2 ,
i i e e
i
n v p
and .
12 2 . 1 2 . 0
2 ,
o o e e
o
n v p
(26,27)
The total operation cost is the next:
).
( )
( , o o,e
o o E e i i i i E
op m p p
c p m p
c
C
(28)
where is cE is a specific operational cost ($/kWh). The total cost is the amount of the material and operational costs:
op m
total C C
C . (29)
2.3. BOUNDARY CONDITIONS
The diameter of inner tube must be at least 25mm. Under this size, the inner velocity will too big value, which occurs a fast erosion in the tube wall. Naturally, the outer tube must be bigger diameter than the inner one. If these two diameters are close together, the velocity in the annulus will be high, this could be also harmful.
The escaping temperature of the outer media must be a higher value than the entering temperature. The flow rate and the tube length must be positive numbers.
3. DESIGN EXAMPLE
3.1. WATER AND WATER MEDIA, TECHNICAL FLUID IN THE INNER TUBE A double pipe heat exchanger must be build, when the technical fluid is water, which flows in the inner tube, the inlet flow rate is 2kg/s, the inlet temperature is 80°C and the outlet temperature is 50°C. The cooling stream is water flows in the annulus and its inlet temperature is 20°C. The media flow counter-current in the first case and parallel current in the second case.
Table 1: Optimized values to a heat exchanger using water to water Counter-current Parallel-current
Inner tube d 73.63mm 75,79mm
Outer tube D 114.28mm 132,59mm
Mass flow mo 2.75 kg/s 4,41 kg/s
Outlet
temperature To,out 41.78 °C 33,61°C
Length L 29.99m 31,92m
Material cost Cm 681.56$ 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 2 1 0 )
( exp( )
m T kg a T a T a T a T a
a m m m m m
eth
T
(30)
K kg T J
a T a T a T a T a a
cTeth m m m m m
exp( 0 1 2 2 3 3 4 4 5 5) 1000
)
( (31)
K m T W a T a T a T a T a
a m m m m m
eth
T( )exp( 0 1 2 2 3 3 4 4 5 5)
(32)
s Pa T
a T a T a T a T a
a m m m m m
eth
T( )exp( 0 1 2 2 3 3 4 4 5 5)103
(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.”
REFERENCES
[1] R. L. Mohanty and S. Bashyam, “Numerical analysis of double pipe heat exchanger using heat transfer augmentation techniques,” vol. 18, no. 124, pp.
337–348, 2015.
[2] R. Kareem, “Optimisation of Double Pipe Helical Tube Heat Exchanger and its Comparison with Straight Double Tube Heat Exchanger,” J. Inst. Eng. Ser. C, 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,”
Ind. Eng. Chem., vol. 28, no. 12, pp. 1429–1435, 1936.
[5] P. K. Swamee, N. Aggarwal, and V. Aggarwal, “Optimum design of double pipe heat exchanger,” vol. 51, pp. 2260–2266, 2008.
[6] G. Fábry, Vegipari gépészek kézikönyve. 1987.
