Thermal design, rating and second law analysis of shell and tube condensers based on Taguchi optimization for waste heat recovery based thermal desalination plants
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Abstract
The present work discusses the design and selection of a shell and tube condenser used in Low Temperature Thermal Desalination (LTTD). To optimize the key geometrical and process parameters of the condenser with multiple parameters and levels, a design of an experiment approach using Taguchi method was chosen. An orthogonal array (OA) of 25 designs was selected for this study. The condenser was designed, analysed using HTRI software and the heat transfer area with respective tube side pressure drop were computed using the same, as these two objective functions determine the capital and running cost of the condenser. There was a complex trade off between the heat transfer area and pressure drop in the analysis, however second law analysis was worked out for determining the optimal heat transfer area vs pressure drop for condensing the required heat load.
Nomenclature
 A
Net cross flow / heat transfer area (m^{2})
 C_{p}
Specific heat (J/kgK)
 C_{gs}
HTRI Shell side flow regime parameter
 d
Diameter of tube
 F
Cross flow correction factor
 f
Friction factor
 f_{is}
Isothermal friction factor
 F_{fb}
B stream flow fraction (≅0.6)
 F_{mr}
Momentum recovery factor
 G
Total mass flux (kg/m^{2}s)
 g
Acceleration due to gravity (m/s^{2})
 h
Heat transfer coefficient (W/m^{2}K)
 K
Thermal conductivity of wall (W/mK)
 L
Tube Length (m)
 m
Total mass flow rate (kg/s)
 N_{t}
Number of tubes
 Pr
Prandtl number
 p_{t}
Tube pitch (m)
 R
Heat capacity ratio
 R_{f}
Fouling resistance (m^{2}K/W)
 R_{lh}
Homogenous liquid volume fraction
 Re
Reynolds number
 S^{.}_{gen}
Entropy generated
 S
Thermal effectiveness
 T
Temperature of Fluid (K)
 U
Overall heat transfer coefficient (W/m^{2}K)
 y
Weight of vapour fraction
 Δp
Pressure drop (kPa)
 ΔT
Temperature difference (K)
 Δp_{mr}
Two phase momentum pressure drop (kPa)
Greek Symbols
 ρ
Density (kg/m^{3})
 μ
Homogenous dynamic Viscosity (Ns/m^{2})
 φ
Temperature profile function
 ∅_{v}^{2}
Ratio of two phase to vapor phase frictional pressure drop
 α
Momentum diffusivity (m^{2}/s)
 ∅_{h}
Physical property correction factor, heat transfer
 ∅_{p}
Physical property correction factor, pressure drop
Suffixes
 c
Cold fluid
 h
Hot fluid
 s
Shell side
 t
Tube side
 sp
Single phase
 w
Wall temperature
 i
In
 o
Out
 l
Liquid
 v
Vapour
 tp
Two phase
 fric
Friction component
 m
Momentum component
1 Introduction
Condensers are heat transfer equipment that are primarily used for the phase change of a fluid from its gaseous to liquid state by cooling. In this process latent heat is given up by the warm side process fluid and transferred to the cold side process fluid respectively ambient on the condenser. They are the first choice of heat exchangers in industry because of well established procedures for design and manufacturing with a wide variety of materials to ensure longevity Kuppan [1]. 90% of heat exchangers that are used in the process industries are of the shell and tube type Lord [2].
National Institute of Ocean Technology (NIOT), Chennai developed the Low Temperature Thermal Desalination (LTTD) technology to utilize the ocean resources in an ecofriendly manner and in order to address the scarcity of drinking water in remote islands NIOT has successfully installed Desalination Plants based on this technology at Kavaratti (2005), Minicoy and Agatti (2011) islands at the Union Territory of Lakshadweep, India.
A baseline model of the shell and tube condenser designed in this work was based on the experience gained by authors in this field through literature survey. The X shell is characterized by pure shell side cross flow and no transverse baffles are used in the X shell; however, support plates are used to suppress the flowinduced vibrations. For a given set of conditions, the X shell has the lowest shell side pressure drop compared to all other shell types (except the K shell). Hence, it is used for cooling applications and for condensing under vacuum [1].
Soltan et al. [5] envisaged that optimization of shell and tube condensers from an economic point of view involves both capital and operating costs and pointed out that the known commercial design procedures and tools do not consist of optimization strategies. Allen et al. [6] suggested that for a given process, the geometry leading to the lowest cost is difficult to determine and complex optimal tradeoffs have to be found.
Yang et al. [7] chose an objective function to minimize the total cost, by dividing a shell and tube heat exchanger into several inseries heat exchangers and simultaneously optimize all the sub divided heat exchangers. Fettaka et al. [8] chose nine decision variables that include the outer diameter of the tube, thickness of the tube, length of tube and baffle spacing.
Selbas et al. [9] used an binarycoded genetic algorithm to minimize the cost and took effective tube pitch, outer diameter of shell, tube outer diameter, number of passes, and baffle cut as their decision variables.
Babu et al. [10] used differential evolution (DE) optimization for the design of heat exchanger and chose the minimization of cost as objective. Caputo et al. [11] employed the MATLAB genetic algorithm toolbox and their objective was the sum of capital cost and discounted annual energy consumption for pumping. Three decision variables namely the shell diameter, tube diameter and baffle spacing was used in their study. Sanaye et al. [12] conducted a multi objective optimization for the estimation of shellside heat transfer coefficient and its corresponding pressure drop. Their results indicated that the length of tube, tube pitch ratio, number of tubes and baffle spacing ratio were responsible for the complex trade off between the total cost and its effectiveness.
Patel et al. [13] envisaged that heat exchanger design can be a complex task and usages of advanced optimization tools are useful to identify the best and cheapest heat exchanger for a specific heat duty. Costa et al. [14] stressed that in spite of all algorithmic developments applied to the heat exchanger design, the complexity of the procedure draws some criticism on the effectiveness of optimization procedures for real industrial problems. Eryener et al. [15] reported that one of the most valuable methods of improving the actual heat transfer coefficient is by the use of a baffle arrangement, however with the effective increase in the heat transfer as a result of decreased flow area there is a steep increase in the shell side pressure drop. In the present study, the baseline design was arrived at based on the TEMA Standards [16].
A condenser that minimizes heat transfer area by varying the geometry can only be achieved only at the expense of pumping power. It is necessary to arrive at such dimensions of a condenser that result under given economic conditions i.e., optimal heat transfer and pressure drop. The results from the literature indicated that optimization of heat exchanger cost was based on minimising the heat transfer area and most works aim towards the concurrent choice of a few configuration design parameters. Few researchers focussed on the impacts of changing a solitary parameter and it was found that optimizing various condensers for a constant heat load has not been concentrated as such.
So far, the concept of ‘design of experiments’ was employed widely for manufacturing applications and the usage of such optimization tool for the design of a shell and tube condenser is the novel contribution in the present work.
2 Design of experiments using Taguchi method
The Taguchi method follows a systematic approach in which the design to be analyzed will be decided first and it provides an optimized design, which has a higher performance index at a lower overall cost. This method has been demonstrated by various researchers to many product development situations, such as Lin et al. [17] and Kang et al. [18] to name a few. According to the standard reference texts by Phadke [19] and Ross [20] the traditional experimental design procedures focus on the average product or process performance characteristics. These authors mentioned that this method concentrates on the effect of variation on the product or process quality characteristics rather than on its averages.
Compared to the full factorial method that works on permutations, Taguchi’s method deals with proper selection of the design parameters during the ‘parametric design’ phase. The approach is based on the principle of minimizing the number of trials by selecting the suitable combination of parameters representing the whole design space, i.e., full matrix of cases. The simultaneous selection and individual assessment of two or more parameters can be done with help of standard orthogonal array.
The parameter design phase of the optimization method applied to the present numerical computations is in accordance to Shaji et al. [21] that includes the following steps: (i) Identify the aim of the analysis; (ii) Identify the quality characteristic and the method of its measurement; (iii) Identify the factors that influence the levels, quality characteristics and interactions that may occur possibly; (iv) Selection of the orthogonal array (OA) that is suitable and factor that need to be assigned at their levels to the particular OA; (v) conduct the numerical analysis as described by the trials in the OA and obtain the responses; (vi) Analysis of the response data by critical examination on the effects created by the factors in the study, the signal to noise ratio and Analysis of Variance to study the factors that were statistically significant to conclude optimum level of factors employed in the study; (vii) Verification of selected optimal design parameters has to be crosschecked with confirmatory analysis. From the seven steps that have been discussed above, the first five steps will be discussed in this numerical study through the following sections.
3 Application of the Taguchi method
The various steps that are involved in the study for the application of the Taguchi method are explained in the following section.
3.1 Identification of the objective for the analysis
 (i)
The first objective was to study the relative influence of various parameters on the overall performance of the shell and tube condenser and the optimal interaction plots results thereof.
 (ii)
The second objective is to identify a suitable case from the given set without compromising on the heat duty by economic calculations.
3.2 Identification of the influencing factors and their levels
Parameters and levels chosen to be analyzed with the optimization algorithm
S.No  Parameter/Factor  Levels of parameters  

1  2  3  4  5  
1  Tube length (m) (A)  8.534  9.754  10.973  12.192  13.411 
2  Tube thickness (mm) (B)  0.559  0.711  0.889  1.245  1.651 
3  Tube outer diameter (mm) (C)  19.05  22.225  25.4  31.75  38.1 
4  No of supports (D)  5  6  7  8  9 
5  Tube side velocity (m/s) (E)  1  1.2  1.4  1.6  1.8 
3.3 Selection of appropriate orthogonal array
In this optimization technique, the parameters are arranged in a specific matrix with the choice of levels where the parametric variations are done called the Orthogonal Array (OA). In a manufacturing standpoint, the process parameters will be varied according to a specific orthogonal array based on the selection and the measured results from the experiments will be fed back to the optimization algorithm, which are referred to as ‘responses’. Presently, the designs and numerical analysis of the shell and tube condensers were developed based on the orthogonal array for the given constant heat load. The estimated values of pressure drop across the tubes and the heat transfer area from the HTRI design analysis form the ‘numerical responses’. Developing the shell and tube condensers based on the chosen parameters, with the chosen levels will result in a large number of designs. A permutation of the identified five numbers of parameters with five levels in each parameter will yield a total number of designs to be analyzed as 3125. Analyzing these many number of designs will be time consuming and computationally expensive. To efficiently manage the number of designs to be analyzed, optimization of the numerical design was employed for the present study. The chosen optimization technique identifies a select set of cases that needs to be analyzed numerically from a large matrix of cases that was originally identified. Since there are five variables with five different levels L25 orthoganal array was selected for this study.
L25 orthogonal array (OA) for the considered parameters and levels
Cases  A  B  C  D  E 

1  8.534  0.559  19.05  5  1 
2  8.534  0.711  22.225  6  1.2 
3  8.534  0.889  25.4  7  1.4 
4  8.534  1.245  31.75  8  1.6 
5  8.534  1.651  38.1  9  1.8 
6  9.754  0.559  22.225  7  1.6 
7  9.754  0.711  25.4  8  1.8 
8  9.754  0.889  31.75  9  1 
9  9.754  1.245  38.1  5  1.2 
10  9.754  1.651  19.05  6  1.4 
11  10.973  0.559  25.4  9  1.2 
12  10.973  0.711  31.75  5  1.4 
13  10.973  0.889  38.1  6  1.6 
14  10.973  1.245  19.05  7  1.8 
15  10.973  1.651  22.225  8  1 
16  12.192  0.559  31.75  6  1.8 
17  12.192  0.711  38.1  7  1 
18  12.192  0.889  19.05  8  1.2 
19  12.192  1.245  22.225  9  1.4 
20  12.192  1.651  25.4  5  1.6 
21  13.411  0.559  38.1  8  1.4 
22  13.411  0.711  19.05  9  1.6 
23  13.411  0.889  22.225  5  1.8 
24  13.411  1.245  25.4  6  1 
25  13.411  1.651  31.75  7  1.2 
4 Design of shell and tube condensers
4.1 Selection of commercial design suite
Computer aided design of equipments forms a vital part of industrial practice, the same applies to the thermal design of components that is required for heat transfer in the process industry known as heat exchangers. There are several wellknown commercial suites that are available for the analysis of heat exchangers in the world with the commonly used ones being developed by Heat Transfer Research, Inc. (HTRI Xchanger Suite software), CHEMCAD software developed by the Chemstations, Inc. and Aspen Shell & Tube Exchanger software by the Aspen Technology, Inc. Paciska et al. [22] investigated the suitability of commercial design suite that is available in the market and validated with the experimental results and suggested that HTRI Xchanger Suite is superior compared to the other two packages for the design of condenser. Aspen Shell & Tube Exchanger performs similarly as CHEMCAD but lacks a large set of functionalities that has been offered by HTRI Xchanger Suite to critically evaluate the thermal design of the equipment.
The Estimation of minimum heat transfer area that is required for a given heat duty is the major objective in any heat exchanger design since it causes escalation in the capital cost of the equipment, Bhatt et al. [23] envisaged that there is no commercial software available in the market that can perform optimization simultaneously with the design and it is also difficult for the designer to justify that the condenser design is the optimised one.
4.2 Heat exchanger calculations used in the simulation
The temperature gradient across the length of heat exchanger during the heat transfer process is non linear for both the fluids, considering arithmetic mean temperature difference (ΔT_{am}) will lead to under estimation of the overall heat transfer area [24] and it leads to incomplete condensation. This results in increase of vacuum pump load which in turn increases the running cost of the system; also improper condensation of generated vapour reduces the quantity and quality of fresh water generated. To avoid this unfavorable effect the actual temperature profile along the path of heat exchanger is identified, to obtain logarithmic mean temperature difference (ΔT_{lmtd}) which gives a better approximation of heat transfer area during cocurrent and counter current flows. In general, cross flow and multi pass heat exchangers the fluid flow direction is not always cocurrent or counter current, this deviation in flow direction results in variation of average driving force. Therefore a correction factor ‘F’ is introduced to determine the actual driving force during the heat transfer process. In condensers and boilers temperature of one of the fluids remain constant therefore the correction factor ‘F’ can be considered as unity, however in this case the condensing fluid experiences subcooling that results in variation of average driving force in the subcooled region. Hence for a conservative design, correction factor was adopted for determining the actual heat transfer area required. For cross flow the correction factor is given by Eq. (2) [25]. Therefore Eq. (1) is rewritten as Eq. (5)
Determination of h_{s} is critical in sizing of the condenser as over and poor estimation will result in insufficient perfomance of the system. Proper understanding of hydrothermal behaviour of hot fluid is necessary as shell side experiences two phase flow condition, making it a complex flow dynamics problem.
The fluctuations in flow patterns and jaunts in flows characterises the hydrodynamic nonequilibrium in most cases for two phase flows to achieve the hydrodynamic equilibrium, several hundreds of diameter of tube distance from the entrance of flow channel is required [27]. Similarly a thermodynamic non equilibrium is characterised by the presence of different phases across various flow patterns. These lack of equipoises are not mutually exclusive as the nonequilibriums may occur simultaneously and may have a profound impact on heat transfer mechanisms. In the existing design procedures for multi component mixture system [28, 29] and systems with non condensable gases [30] it is assumed that the phases are to be in equilibrium. A large interface area and a high degree of turbulence is required to achieve a quasi equilibrium state between two phases which is hard to accompolish. Thus accurate determination of the flow regime is critical to determine the governing heat transfer mechanisms in the flow direction.
The single phase heat transfer coefficient (h_{t}) is calculated by Gnielinski’s correlation & Wilson plot method, three equation model [37].
Where a_{f} = 0.0035, b_{f} = 0.264, C_{f} = − 0.42 are constants that take into account the impact of surface roughness on friction factor [34]. The equations that were cited above are from the literature used in the commercial suite.
4.3 Determination of Thermodynamic Irreversibility and Strength of Convection Current
Be_{ther}=1 corresponds to the existence of irreversibility only due to heat transfer. Similarly a Bejan number Be_{ther} ≈0 corresponds to cases in which the irreversibility is dominated by fluid friction. Hence a higher value of Be_{ther} indicates that a system experience minimum irreversibility due to fluid flow friction. In case of heat exchangers it has been observed that in most cases irreversibilities due to the heat transfer process has less impact on total entropy generation rate than the irreversibility from the fluid friction component [42]. Higher value of Bejan number (Be_{ther}) makes the system thermodynamically efficient.
Higher values of Be_{ther} & Be_{HT} number indicate optimum performance of a heat exchanger.
4.4 Process considerations in the simulations
Kailasam [43] investigated and reported that thermal effluent discharge at the Tuticorin bay was 38.92 °C, so in this case 39 °C was selected for the study as the warm fluid inlet temperature and 30 °C as the cold fluid inlet temperature, since the temperature varied from 28 to 30 °C, a conservative value was chosen to acheive the desired output even at deprived conditions. The formation of undesired deposits on heat transfer surfaces that obstructs the effective heat transfer and increases the resistance to fluid flow is known as fouling. The growth of the deposits causes the thermo hydraulic performance of heat exchanger to degrade with time.
Composition of the warm fluid
S.No.  Component  Weight fraction 

1  Water vapour  0.9969 
2  Nitrogen  0.0019 
3  Oxygen  0.0011 
4  Argon  5.127e5 
5  Carbon di oxide  4.217e5 
6  Neon  3.176e8 
7  Helium3  7.254e9 
8  Hydrogen  1.529e9 
9  Carbon monoxide  7.45e10 
10  Air  1.084e4 
Total  1 
 1.
The plant capacity of 1 Million Litres per Day
 2.The fouling resistances

Shell side – 8.8E05 m^{2}K/W

Tube side – 1.32E04 m^{2}K/W

 3.
Rotated square tube layout for all designs
 4.
CuNi 90/10 tubes for all designs
 5.
Excess generation of 3% (1.03MLD)
 6.
Uncondensed watervapour in condenser as 2%
 7.
Minimum of 5% excess heat transfer for all designs
The assumptions are made, keeping in mind the 1MLD fresh water has to be generated even in deprived conditions.
With the above constraints and with the given set of parameters for each case, the condensers were designed.
5 Results and discussions
5.1 HTRI results
The Fig. 3b scaled down to 1:50 shows that AXL type heat exchanger was used with the shell inner diameter of 4300 mm and tube OD of 19.05 mm with tube pitch of 25.51 mm along with rotated square layout.
Results obtained from HTRI
Cases  Heat transfer area (m^{2})  Tube side Pressure drop (kPa) 

1  8337.6  9.78 
2  6765.3  9.6 
3  5804.0  12.38 
4  5302.1  13.88 
5  5211.9  15.18 
6  5559.7  20.27 
7  5172.8  20.13 
8  6891.1  6.63 
9  5992.2  8.07 
10  7633.2  21.82 
11  6251.6  11.72 
12  5688.7  10.86 
13  5099.5  12.71 
14  6344.8  35.55 
15  8779.8  11.38 
16  5041.2  19.39 
17  6378.5  6.5 
18  8157.1  18.43 
19  6960.6  21.04 
20  6299.4  22.99 
21  5411.4  12.51 
22  6927.9  31.84 
23  6046.6  33.02 
24  8197.6  11.25 
25  6151.6  12.15 
5.2 Second law analysis
Second law analysis results
Cases  Heat transfer rate for each tube (Q), W  Heat transfer coefficient inside (h_{i}), W/m^{2}K  Nusselt number (Nu)  Bejan number (Be_{ther})  Bejan number (Be_{HT} *10^{9}) 

Case9  5541.08  4834.4  277.59  0.64  2.05 
Case12  5482.28  5318.2  260.08  0.57  4.68 
Case13  7325.91  5815.9  340.71  0.50  4.73 
Case16  6886.46  6635.6  327.84  0.38  9.89 
Case17  6508.44  4214.3  249.31  0.73  2.79 
Case21  8439.32  5487.2  327.32  0.53  6.67 
5.3 Interaction plots
For better interpretation of the results and to visualize the corresponding effects due to independent factors, a plot or graph is used. When the number of independent parameters increase it is difficult to explain their effect on the output due to the independent effects on each of the variables. Hence it seems prudent to use interaction plots for observing the combined effects of all the parameters. The interaction plots are selected against the conventional graphs since they depict much better relation/influence of multi parameters on the objective. Interaction plots are generally used for visualizing the possible interactions or dependencies between variables (factors) in designed experiments and represent the combined effects of the variables on the dependent measure. The parallel lines in the plots indicate that there is no significant interaction between the variables and there is an higher degree of interaction when there is greater difference in slope between them.
The interaction of parameters on the heat transfer area as shown in Fig. 6 depicts that interactions between all the parameters that exists are mostly significant except Tube Outer Diameter (C) Vs Tube Side Velocity (E) interaction, since they show a disordinal interaction between the parameters. Thus, almost all the factors are interdependent on each other. Thus it can be concluded that before changing the magnitude of one factor, the interactions between other factors must also been considered because they can significantly affect the response on the output.
The interaction of parameters on the tube side pressure drop as shown in Fig. 7 explains that there is significant interactions between all the parameters that have been used in this study since a large degree of interactions are seen from the plots. It is interesting to observe that the interactions between the parameters Tube side velocity (E) Vs Tube outer diameter (C) do not show a greater degree of interaction, but there is possible interactions between the parameters since the lines are disordinal. Thus it can be concluded that all factors are interdependent on each other and a change in individual parameter requires a change in all the parameters.
Although interaction plots are useful in identifying the degree of interactions between the variables, those contribute to fiveway interaction. The significant differences among any two, three or all four combinations of the selected variables in the higher and lower levels can be estimated through statistical analysis, which is being taken up as a next phase of this work and also to develop correlations for the output variables with respect to the input parameters.
5.4 Discussions
The condenser has to condense a capacity of 1.03 million litres per day (MLD), so for the given capacity it would be ideal to choose a condenser that offers lowest heat transfer area and tube side pressure drop.
The heat transfer area of 5041.2 m^{2} in case 16, was the lowest for the given heat load and the pressure drop on the tube side was observed to be 6.50 kPa in case 17. Similarly Case 15 and Case 14 reported the highest heat transfer area and tube side pressure drop of 8779.8 m^{2} and 35.55 kPa respectively. There was a trade off between the heat transfer area and pressure drop in the analysis however based on the second law analysis it was concluded Case 21 had the optimal heat transfer area vs pressure drop with the values of 5411.4 m^{2}and 12.51 kPa respectively.
6 Conclusions
The present work discusses the thermal design and rating of shell and tube condensers using the Taguchi approach. The design modifications were carried out by varying the geometrical parameters with various levels. The heat transfer area and the pressure drop across the tube were estimated, compared and reported. There was a trade off between the heat transfer area and pressure drop in the analysis however it is concluded that Case 21 had the optimal heat transfer area vs pressure drop based on the second law analysis. Results need to be analysed with parameters taken individually to find the most significant one. Hence, the future work will be done taking into account the Heat transfer area and Tube side pressure drop as objective functions and finding the significance of the results through ANOVA and other numerical analysis that include Regression analysis for identifying the influential parameters in this study.
Notes
Acknowledgments
This work has been done under the funding of Ministry of Earth Sciences (MoES), Govt. of India.
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