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Heat and Mass Transfer

, Volume 54, Issue 9, pp 2885–2897 | Cite as

Thermal design, rating and second law analysis of shell and tube condensers based on Taguchi optimization for waste heat recovery based thermal desalination plants

  • Balaji Chandrakanth
  • Venkatesan G
  • Prakash Kumar L.S.S
  • Purnima Jalihal
  • Iniyan S
Original
  • 98 Downloads

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 (m2)

Cp

Specific heat (J/kgK)

Cgs

HTRI Shell side flow regime parameter

d

Diameter of tube

F

Cross flow correction factor

f

Friction factor

fis

Isothermal friction factor

Ffb

B stream flow fraction (≅0.6)

Fmr

Momentum recovery factor

G

Total mass flux (kg/m2s)

g

Acceleration due to gravity (m/s2)

h

Heat transfer coefficient (W/m2K)

K

Thermal conductivity of wall (W/mK)

L

Tube Length (m)

m

Total mass flow rate (kg/s)

Nt

Number of tubes

Pr

Prandtl number

pt

Tube pitch (m)

R

Heat capacity ratio

Rf

Fouling resistance (m2K/W)

Rlh

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/m2K)

y

Weight of vapour fraction

Δp

Pressure drop (kPa)

ΔT

Temperature difference (K)

Δpmr

Two phase momentum pressure drop (kPa)

Greek Symbols

ρ

Density (kg/m3)

μ

Homogenous dynamic Viscosity (Ns/m2)

φ

Temperature profile function

v2

Ratio of two phase to vapor phase frictional pressure drop

α

Momentum diffusivity (m2/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 eco-friendly 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.

This technology can also generate fresh water by utilizing the low grade waste heat energy from thermal power plants, which in turn helps to reduce the thermal pollution. The line diagram of low temperature thermal desalination plant is as shown in Fig. 1. Flash Chamber is a pressure vessel operated with an internal pressure of about 55 mbar (abs) and the external atmospheric pressure that has been used to evaporate the water available from the power plant condenser outfall as waste heat at 39 °C under vacuum conditions. The generated vapour is then condensed in Shell and Tube condenser. The surface sea water at a temperature of 30 °C is pumped to the condenser as the cold working fluid on the tube side. The vacuum system is connected to the shell side of the condenser to maintain the low pressure in the system that is less than the saturation pressure of the warm water [3, 4].
Fig. 1

Schematic diagram of low temperature thermal desalination plant

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 flow-induced 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 trade-offs 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 in-series 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 binary-coded 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 shell-side 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

The objectives of the present numerical analysis were two fold as described below.
  1. (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.

     
  2. (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

The factors that influence the performance of a shell and tube condenser were the geometry parameters and certain operating parameters that dictate the design of a particular condenser. For the purpose of analyzing the shell and tube condenser, key parameters were identified from exhaustive literature survey and the sample range was chosen based on standards [16]. The key parameters identified for studying the relative influence are Tube length (m), Tube thickness (mm), Tube Outer diameter (mm), Number of supports and Tube side velocity (m/s). The identified parameters with the considered levels are furnished in Table 1.
Table 1

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.

Minitab software was used for the optimization where the Taguchi algorithm, an option within the software was selected for identifying the orthogonal array from the input matrix of parameters with chosen levels. An orthogonal array (OA) of 25 cases (L25) for 5 parameters with 5 levels of variations was identified from the optimization tool. The L25 orthogonal array of the different parameters that has been selected for this study is as shown in Table 2. The two responses ‘heat transfer area’ and ‘pressure drop across the tubes’ of the shell and tube condenser were considered, these responses needs to be minimized since it involves the capital cost and the running cost of the condenser.
Table 2

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 well-known 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 difference between source and sink acts as the driving force for any heat transfer process. In a heat exchanger the role of source and sink are performed by hot and cold fluid respectively. The amount of heat transferred in the heat exchanger is given by
$$ \mathrm{Q}=\mathrm{UA}\left({\Delta \mathrm{T}}_{\mathrm{am}}\right) $$
(1)

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 (ΔTam) 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 (ΔTlmtd) which gives a better approximation of heat transfer area during co-current and counter current flows. In general, cross flow and multi pass heat exchangers the fluid flow direction is not always co-current 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)

$$ \mathrm{F}=\frac{\sqrt{\left({\mathrm{R}}^2+1\right)}\ln \left(\frac{1-\mathrm{S}}{1-\mathrm{RS}}\right)}{\left(\mathrm{R}-1\right)\ln \left[\frac{2-\mathrm{S}\left(\mathrm{R}+1-\sqrt{{\mathrm{R}}^2+1}\right)}{2-\mathrm{S}\left(\mathrm{R}+1+\sqrt{{\mathrm{R}}^2+1}\right)}\right]} $$
(2)
$$ \mathrm{R}=\frac{{\mathrm{T}}_{\mathrm{h},\mathrm{i}}-{\mathrm{T}}_{\mathrm{h},\mathrm{o}}}{{\mathrm{T}}_{\mathrm{c},\mathrm{o}}-{\mathrm{T}}_{\mathrm{c},\mathrm{i}}} $$
(3)
$$ \mathrm{S}=\frac{{\mathrm{T}}_{\mathrm{c},\mathrm{o}}-{\mathrm{T}}_{\mathrm{c},\mathrm{i}}}{{\mathrm{T}}_{\mathrm{h},\mathrm{i}}-{\mathrm{T}}_{\mathrm{c},\mathrm{i}}} $$
(4)
$$ \mathrm{Q}=\mathrm{FUA}\left({\Delta \mathrm{T}}_{\mathrm{lmtd}}\right) $$
(5)
The overall heat transfer coefficient is a function of both shell and tube side heat transfer coefficients and the flow resistance caused due to the fouling of heat exchanger tubes [11];
$$ \mathrm{U}=\frac{1}{\frac{1}{{\mathrm{h}}_{\mathrm{s}}}+{\mathrm{R}}_{\mathrm{fs}}+\frac{{\mathrm{d}}_{\mathrm{o}}}{{\mathrm{d}}_{\mathrm{i}}}\left({\mathrm{R}}_{\mathrm{ft}}+\frac{1}{{\mathrm{h}}_{\mathrm{t}}}\right)+\frac{{\mathrm{d}}_{\mathrm{o}}\ln \left({\mathrm{d}}_{\mathrm{o}}/{\mathrm{d}}_{\mathrm{i}}\right)}{2{\mathrm{k}}_{\mathrm{t}}}} $$
(6)
Sizing of the heat exchanger is carried out by determining the actual heat duty at which the heat exchanger is being operated. The heat duty of the heat exchanger is given by Kern [26] using a simple energy balance equation, where ΔTc = Tc, o − Tc, i and ΔTh = Th, i − Th, o and these differences are computed as modulus.
$$ \mathrm{Q}={\mathrm{m}}_{\mathrm{c}}{\mathrm{C}}_{\mathrm{pc}}{\Delta \mathrm{T}}_{\mathrm{c}}={\mathrm{m}}_{\mathrm{h}}{\mathrm{C}}_{\mathrm{ph}}{\Delta \mathrm{T}}_{\mathrm{h}} $$
(7)

Determination of hs 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 non-equilibrium 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 non-equilibriums 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 flow patterns across the shell side of a condenser can be grouped into two major flow regimes: Gravity controlled flow and Shear controlled flow. In horizontal flow the determination of flow regime is done using Baker’s flow regime map [31]. However baker’s correlation does not hold good agreement for high pressure, stratified flow and low surface tension fluids [32, 33]. Therefore in horizontal tube three parameters are used to determine the flow regime: HTRI shell side flow regime parameter (Cgs), Total mass Velocity (Gs), homogenous flow liquid volume fraction (Rlh) as given in Eqs. (8), (9), and (10).
$$ {\mathrm{C}}_{\mathrm{gs}}=\frac{1}{{\mathrm{G}}_{\mathrm{t}}}{\left[\left({\mathrm{p}}_{\mathrm{t}}-{\mathrm{d}}_{\mathrm{o}}\right)\mathrm{g}{\uprho}_{\mathrm{v}}\left({\uprho}_{\mathrm{l}}-{\uprho}_{\mathrm{v}}\right)\left(\frac{1-\mathrm{y}}{\mathrm{y}}\right)\right]}^{0.5} $$
(8)
$$ {\mathrm{G}}_{\mathrm{s}}=\frac{{\mathrm{m}}_{\mathrm{t}}{\mathrm{F}}_{\mathrm{fb}}}{{\mathrm{A}}_{\mathrm{s}}} $$
(9)
$$ {\mathrm{R}}_{\mathrm{l}\mathrm{h}}=\frac{1}{1+\left(\frac{\mathrm{y}}{1-\mathrm{y}}\right)\frac{\uprho_{\mathrm{l}}}{\uprho_{\mathrm{v}}}} $$
(10)
The condensed fluid slides over the tube and settles at bottom of the condenser (gravity controlled) if the flow regime parameter Cgs is greater than 0.7, similarly the condensed fluid is carried by vapour in flow direction (shear controlled) if Cgs is less than 0.3. The shell side flow regimes are estimated from Fig. 2 using Rlh and Cgs calculated from Eqs. (8) and (9). The value of B stream flow fraction (Ffb) was given by Yang [35].
Fig. 2

HTRI generalised flow regime map for Horizontal flows [34]

Bell k J [36] envisaged that a detailed design approach incorporating the effect of laminar and turbulent nature of flow along with different flow regimes is required. The governing equations for determining the heat transfer coefficient for laminar (Eq. 11) and turbulent region (Eq. 12) is given below.
$$ {\mathrm{h}}_{\mathrm{s}}=1.6{\mathrm{C}}_{\mathrm{l}\mathrm{r}}{\mathrm{k}}_{\mathrm{l}}{\left[\frac{\uprho_{\mathrm{l}}\left({\uprho}_{\mathrm{l}}-{\uprho}_{\mathrm{v}}\right)\mathrm{g}}{{\upmu_{\mathrm{l}}}^2}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left({\operatorname{Re}}_{\mathrm{v}}\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} $$
(11)
$$ {\mathrm{h}}_{\mathrm{s}}=0.015{\mathrm{k}}_{\mathrm{l}}{\left[\frac{\uprho_{\mathrm{l}}\left({\uprho}_{\mathrm{l}}-{\uprho}_{\mathrm{v}}\right)\mathrm{g}}{{\upmu_{\mathrm{l}}}^2}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left({\operatorname{Re}}_{\mathrm{v}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\left({\Pr}_{\mathrm{l}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} $$
(12)
Where Clr is the laminar film rippling factor [30] based on Reynolds’s number and given by Eqs. (13)–(15)
$$ {\operatorname{Re}}_{\mathrm{v}}\le 40,{\mathrm{C}}_{\mathrm{lr}}=1.0 $$
(13)
$$ {\operatorname{Re}}_{\mathrm{v}}\ge 200,{\mathrm{C}}_{\mathrm{lr}}=1.5 $$
(14)
$$ 40<{\operatorname{Re}}_{\mathrm{v}}<200,{\mathrm{C}}_{\mathrm{lr}}=0.003 \operatorname {Re}+0.875 $$
(15)
The total pressure drop across the shell side is the sum of pressure drop in the de superheating and condensing zones. Total pressure drop in condensing zone consists of friction and momentum components as given below.
$$ {\Delta \mathrm{p}}_{\mathrm{tp}}={\Delta \mathrm{p}}_{\mathrm{fric}}+{\Delta \mathrm{p}}_{\mathrm{m}} $$
(16)

The single phase heat transfer coefficient (ht) is calculated by Gnielinski’s correlation & Wilson plot method, three equation model [37].

This model holds well only if there are two constants and one variable flow parameter to be determined from the equations. In general the thermal resistance of the flow is taken as the variable in determining the heat transfer coefficient, however at elevated temperatures the thermal resistance of water becomes almost constant [38]. Kakac et al. [39] tested Gnielinski’s correlation at higher Reynolds number, temperature and suggested a modified equation as given below showed better accuracy.
$$ \mathrm{Nu}=\frac{\mathrm{f}}{2}\mathrm{RePr}\upvarphi {\varnothing}_{\mathrm{h}} $$
(17)
$$ \mathrm{RePr}\upvarphi =\frac{\left(\operatorname{Re}-1000\right)\Pr }{1+12.7{\left(\frac{\mathrm{f}}{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({\mathrm{pr}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-1\right)} $$
(18)
$$ \mathrm{f}={\left(1.58\ln \left(\operatorname{Re}\right)-3.28\right)}^{-2} $$
(19)
$$ {\varnothing}_{\mathrm{h}}={\left(\frac{\upmu}{\upmu_{\mathrm{w}}}\right)}^{0.11} $$
(20)
In the tubeside, pressure drop is mainly due to friction while pressure drop due to momentum and static head is very less and can be neglected. Thus the pressure drop is calculated using Eq. (21).
$$ {\Delta \mathrm{p}}_{\mathrm{sp}}=\left[4\mathrm{f}\left(\frac{\mathrm{l}}{{\mathrm{d}}_{\mathrm{i}}}\right)+{\mathrm{K}}_{\mathrm{v}}\right]\left(\frac{{{\mathrm{G}}_{\mathrm{t}}}^2}{2\uprho {\mathrm{g}}_{\mathrm{c}}}\right) $$
(21)
$$ \mathrm{f}={\mathrm{f}}_{\mathrm{is}}{\varnothing}_{\mathrm{p}} $$
(22)
$$ {\mathrm{f}}_{\mathrm{is}}={\mathrm{a}}_{\mathrm{f}}+{\mathrm{b}}_{\mathrm{f}}{\operatorname{Re}}^{{\mathrm{c}}_{\mathrm{f}}} $$
(23)

Where af = 0.0035, bf = 0.264, Cf =  − 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

Some of the important parameters that define the heat transfer and flow characteristic of a heat transfer equipment are fluid throttling, fluid flow friction, phase change and the actual heat transfer at finite temperature difference. Thermodynamic analysis of these parameters indicate the irreversibility that occurs during the process. An indicator of these process irreversibilities is entropy generation within the system which is a major cause of deterioration of heat exchanger performance. The losses in heat exchanger performance due to entropy generation can be calculated using the second law analysis. Entropy generated in a heat exchanger is mainly due to the fluid flow friction and heat transfer due to finite temperature difference. Hence total entropy generated in the system is the summation of these two parameters and given in Eq. (24)
$$ {{\mathrm{S}}^{.}}_{\mathrm{gen}}={{\mathrm{S}}^{.}}_{\mathrm{gen},\Delta \mathrm{T}}+{{\mathrm{S}}^{.}}_{\mathrm{gen},\Delta \mathrm{P}} $$
(24)
Where S.gen, ∆T is the entropy generation due to heat transfer and S.gen, ∆P is the entropy generation due to fluid flow friction. Second law analysis of tubular heat exchanger with constant heat flux was presented by Zimparov [40], the entropy generation rate is given as
$$ {{\mathrm{S}}^{.}}_{\mathrm{gen}}=\frac{{\mathrm{Q}}^2}{{{\mathrm{N}}_{\mathrm{t}}}^2{{\mathrm{T}}_{\mathrm{i}}}^2\uppi {\mathrm{k}}_{\mathrm{f}}\mathrm{NuL}}\frac{1}{\left(1+\raisebox{1ex}{${\Delta \mathrm{T}}_{\mathrm{m}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{T}}_{\mathrm{i}}$}\right.\right)}+\frac{32{\mathrm{m}}^3\mathrm{fL}}{{{\mathrm{N}}_{\mathrm{t}}}^3{\uprho}^2{\uppi}^2{\mathrm{T}}_{\mathrm{i}}{\mathrm{d}}^5}\ \frac{\ln \Big(1+\raisebox{1ex}{${\Delta \mathrm{T}}_{\mathrm{m}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{T}}_{\mathrm{i}}\Big)$}\right.}{\left(1+\raisebox{1ex}{${\Delta \mathrm{T}}_{\mathrm{m}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{T}}_{\mathrm{i}}$}\right.\right)} $$
(25)
The first two terms on the right hand side of the said equation accounts for the entropy generation due to heat transfer and fluid flow friction respectively. The relative importance of these irreversibilities were investigated by Natalini and Sciubba [41] and termed as Bejan number. Bejan number (Bether) is the ratio of irreversibility due to heat transfer to the total irreversibility given by Eq. (26).
$$ {\left(\mathrm{Be}\right)}_{\mathrm{ther}}=\frac{{{\mathrm{S}}^{.}}_{\mathrm{gen},\Delta \mathrm{T}}}{{{\mathrm{S}}^{.}}_{\mathrm{gen},\Delta \mathrm{T}}+{{\mathrm{S}}^{.}}_{\mathrm{gen},\Delta \mathrm{P}}} $$
(26)

  Bether=1 corresponds to the existence of irreversibility only due to heat transfer. Similarly a Bejan number Bether ≈0 corresponds to cases in which the irreversibility is dominated by fluid friction. Hence a higher value of Bether 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 (Bether) makes the system thermodynamically efficient.

However not all the systems which are thermodynamically efficient can be optimal in terms of the heat transfer rate. This is due to the fact that a thermodynamically efficient heat exchanger has lowest irreversibility due to fluid flow friction which corresponds to low flow velocity. This reduced velocity of fluid flow causes a severe degradation of heat transfer in the system as heat transfer is primarily due to conduction than convection. In forced convection, Bejan number (BeHT) given in Eq. (27) is used to determine the dominant mode of heat transfer by grouping a dimensionless number that indicates the thickness of thermal boundary layer [43]. Higher values of BeHT indicate stronger convection currents in the flow thus resulting in better heat transfer rate.
$$ {\left(\mathrm{Be}\right)}_{\mathrm{HT}}=\frac{\Delta \mathrm{P}{\mathrm{L}}^2}{\upmu \upalpha} $$
(27)

Higher values of Bether & BeHT number indicate optimum performance of a heat exchanger.

4.4 Process considerations in the simulations

All phase changes take place at a specific temperature and for a given pressure, which involves only change in internal potential energy. The mass flow rate in Eq. (28) shows the actual quantity of the fresh water generated per day. The warm water is obtained from the power plant condenser outfall which is the waste heat energy discharged to the sea. The specific latent heat (LH) of a material is a measure of the heat energy (Q) per mass (mf) released or absorbed during a phase change and it is given by the Q = mf .LH, here the mf quantifies the amount of water vapour which enters through the condenser and corresponding latent heat has to be done.
$$ Mass\ flow\ rate=\frac{plant\ capacity}{Duration} $$
(28)

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.

The effect of Non-Condensable gases (NC) has to be considered during condensation since there are some insoluble gases that are liberated from feed seawater during flashing which acts as the additional load to the condenser. The percentage contributions of the gases as shown in Table 3 are also considered in the analysis to ensure that the condenser is designed to withstand the additional load.
Table 3

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

5.127e-5

5

Carbon di oxide

4.217e-5

6

Neon

3.176e-8

7

Helium-3

7.254e-9

8

Hydrogen

1.529e-9

9

Carbon monoxide

7.45e-10

10

Air

1.084e-4

 

Total

1

Assumptions & Constraints considered in the analysis
  1. 1.

    The plant capacity of 1 Million Litres per Day

     
  2. 2.
    The fouling resistances
    • Shell side – 8.8E-05 m2-K/W

    • Tube side – 1.32E-04 m2-K/W

     
  3. 3.

    Rotated square tube layout for all designs

     
  4. 4.

    Cu-Ni 90/10 tubes for all designs

     
  5. 5.

    Excess generation of 3% (1.03MLD)

     
  6. 6.

    Uncondensed watervapour in condenser as 2%

     
  7. 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.

For 1 MLD capacity, the mass flow rate of the water vapour has to be
$$ Mass\ flow\ rate=\frac{1000000}{24\ast 3600}=11.57\ kg/s $$

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

It is difficult to compare and show all the condensers in the design, it is better to analyse any one condenser since all the condensers are designed for the same heat loads hence The Fig. 3a shows the geometrical layout of the condenser, it can be observed that the no. of support plates used was 5 along with 3 warmfluid inlet ports and 2 freshwater outlet ports.
Fig. 3

a Geometry of Shell and tube condenser. b Condenser Layout

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.

The total tubes of 16,647 used in case 1 condenser arranged in 2 rows with the first row holding 8344 and second row with 8344 tubes respectively. The Fig. 3b also shows that there is 2 rows of inpingement rods with 19.05 mm to avoid possible erosion and failure of tubes placed at a height of 1007 mm under the warmfluid inlet nozzle. The warmfluid inlet nozzle diameter of 1700 mm along with freshwater port diameter of 250 mm. The results of the 25 cases are furnished in Table 4.
Table 4

Results obtained from HTRI

Cases

Heat transfer area (m2)

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

The decision variables that have been considered in this study, heat transfer area and the tube side pressure drop show a complex tradeoff, so it is therefore required to find the optimal solutions from the given set of trials. Pareto optimal trade off is used in this analysis to find the solutions that are potentially optimal solutions [44]. Figure 4 shows the Pareto optimal solution for the heat transfer area and tube side pressure drop that is required for 25 cases and it was found that Case 16, 13, 21, 12, 9and 17 are the optimal cases. To arrive at the unique optimal case, second law analysiswere carried out for the optimal cases arrived from Pareto optimal trade off and the corresponding results are furnished in Table 5.
Fig. 4

Heat transfer area and Tube side pressure drop required for 25 cases with Pareto-optimal solutions

Table 5

Second law analysis results

Cases

Heat transfer rate for each tube (Q), W

Heat transfer coefficient inside (hi), W/m2K

Nusselt number (Nu)

Bejan number

(Bether)

Bejan number

(BeHT *109)

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

Figure 5 shows the second law comparison of 6 shell-and-tube heat exchanger cases that has been obtained from the Pareto optimal solutions. The shell and tube heat exchanger design based on case16 indicated a maximum BeHT of 0.9*1010 out of these cases, however it showed only 0.38 as Bether. Similarly it was observed a minimum BeHT of 0.27*1010 in case 17 but it recorded a maximum of Bether value of 0.73. Since Bether and BeHT varies inversly to each other case 16 & 17 is not optimal. Further investigation on Fig. 7 and Table 5 shows that case 21 with BeHT of 0.67*1010 and Bether of 0.53 as optimal one, since it showed highest effective heat transfer on each tube with lower process irreversibility.
Fig. 5

Convective current vs strength of irreversibility for Pareto optimal cases

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.

Figures 6 and 7, shows the interaction plot with objective of heat transfer area and tube side pressure drop. These graphs demonstrate the behaviour of individual variables on the objective parameters.
Fig. 6

Interaction plot with respect to heat transfer area

Fig. 7

Interaction plot with respect to Tube side Pressure drop

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 inter-dependent 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 five-way 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 m2 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 m2 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 m2and 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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Balaji Chandrakanth
    • 1
  • Venkatesan G
    • 1
  • Prakash Kumar L.S.S
    • 1
  • Purnima Jalihal
    • 1
  • Iniyan S
    • 2
  1. 1.Energy& Fresh Water groupNational Institute of Ocean TechnologyChennaiIndia
  2. 2.Institute of Energy Studies, College of Engineering, GuindyChennaiIndia

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