Optimal design for a shell-tube heat exchanger of a binary geothermal power plant from economic point of view

Abstract

The goals of this study are to design and optimize heat exchangers for 1 MW binary geothermal power system. It comprises two components i.e., preheater and evaporator. This study adopts genetic algorithm (GA) to reach an optimized outcome for both capital investment and operating expense aspects. Indeed the use of optimization tools in designing heat exchangers is not new. However, the material specification from Tubular Exchanger Manufacturers Association (TEMA) standards is not always considered in the previous studies. In this case, the design is not practical in actual application due to the availability of the materials and components size. Motivated by this issue, this study further investigates the design with and without TEMA. The design considers various design variables, i.e., tube diameter, number of tube passes, baffle spacing and tube length. A case study is further analyzed to validate the practicality of proposal model. Implications of the results are analyzed and discussed. The results show that total cost for both preheater and evaporator decrease 46.1% and 56.4%, respectively when it is compared to traditional approach without optimization tools.

Keywords

Shell-tube heat exchanger optimal design 1 MW geothermal system genetic algorithm economical design

1. Introduction

Heat exchanger (HE) is a device that transfers heat energy between two or more working fluids at different temperatures. It plays important roles in industrial processes and power plants. As an example, HE is commonly used as evaporator, condenser, cooling tower and etc. for power plants applications. It is worth-mentioning that shell-tube heat exchanger (STHE) is one of the most widely and popular used design for HE. STHE has several advantages, such as ease of maintenance, simple manufacturing, high pressure working ability and adaptability to different operating conditions [1]. Though STHE offers various advantages, its optimized design that reduces the cost and energy is challenging. The design is a complicated task that requires a thorough understanding of mechanical, thermos-dynamics, fluid mechanics, thermos-economics and optimization.

Conventionally, the design of STHE is more toward experience-based design approach that involves numbers of parameters and geometries to be considered. Besides, the design also needs to satisfy a given heat duty requirement and a set of geometric and given operational constraints. The approach is time-consuming and does not guarantee an optimal solution [2]. Recently, a number of researches have been conducted to improve the conventional approach. These researches adopted the optimization tools to enhance the design process in term of computational and cost effectiveness. For examples, there are some researches emphasized on minimizing the entropy generation rate [3] and also others attempt on minimizing the heat exchanger area [4] and/or total cost of heat exchanger [5, 6, 7, 8].

Literature revealed that the optimization algorithms pertaining to optimize the heat exchangers were teach-ing-learning based optimization (TLBO) [2], genetic algorithm (GA) [4, 6, 9, 10, 11, 12, 13], imperialist competitive algorithm (ICA) [7], biogeography-based optimization (BBO) [8, 14], particle swarm optimization (PSO) [1, 15, 16], harmony search optimization (HSO) [17], cu-ckoo search (CS) [18, 19], and artificial bee colony (ABC) [20]. These algorithms have been successfully implemented to design STHE and real case studies have been reported. Though many researches pertaining to STHE have been reported, Tubular Exchanger Manufacturers Association (TEMA) standards are not always considered in the design. TEMA is an association of manufacturers of shell and tube heat exchangers which has established a set of construction standards for STHE. Consequently, the design is ideal but it does not sound practical in real situation as manufacturers may not be able to produce and supply the materials with user-center dimensions and specifications.

The purpose of this study is to adopt optimization tool in designing for a 1 MW binary geothermal system. This study investigates the design with TEMA and without TEMA. Among many optimization tools from the literature, GA is adopted in this study due to its stability, consistency practicality and popularity in various disciplines including economic [21, 22, 23], finance [24, 25], scheduling task [26, 27], manufacturing [28], transportation [29, 30], engineering [31, 32] and etc. For examples, Elliston et al. [21] utilized GA with an existing simulation tool to identify the lowest cost scenarios of renewable technologies and locations for Australian National Electricity Market. López-Lezama et al. [22] applied GA to determine the location and contract pricing of distributed generation units in distribution systems. Mohamed and Koivo [23] used GA to determine the optimal operating strategy and cost optimization scheme for a MicroGrid for residential application. They minimized the cost function of the system while constraining it to meet the customer demand and safety of the system. Acosta-González and Fernández-Rodríguez [24] applied to forecast financial failure of firms. Straßburg et al. [25] employed GA to find the best trading rules in the field of technical analysis of stock markets. Zamani [26] presented an innovated GA for solving the resource-constrained project scheduling problem. The approach used a magnet-based crossover operator which aims to preserve to two contiguous parts from the receiver and one contiguous part from the donator genotype. Xu et al. [27] applied GA for task scheduling on heterogeneous computing systems. Kia et al. [28] utilized GA-based model for a multi-floor layout design of cellular manufacturing systems. The aims are to minimize the total costs of intra-cell, inter-cell, and inter-floor material handling, purchasing machines, machine processing, machine overhead, and machine relocation in the cellular manufacturing systems. Lin et al. [29] proposed a GA-based optimization model for designing a green transportation scheme which is capable of suggesting guidance for the logistics service providers in order to achieve a low economic and environmental cost. Recently, a study is conducted to evaluate the performance of GA and other optimization algorithms in solving a set of 14 well-known engineering optimization problems [32]. The results have shown that the proposed GA exhibits a superior performance in comparison to other algorithms that also solved those problems. In short, GA-based optimization is a promising solution pertaining to engineering problems.

To design for the 1 MW binary geothermal system, preheater and evaporator are two essential components that need to be considered. The designs of these two components are based on the concept of STHE. Numbers of important parameters, such as the tube diameter, the number of tube passes, the baffle spacing the tube length and etc., are needed to be considered and hence it is a complex combinatory problem in which the need of GA is essential. To validate the performance of the proposed GA-based model, a series of experiment is conducted. First of all, a benchmarked example is adopted and the performance of the GA-based model is compared to the conventional design, i.e., without the use of optimization tools. Subsequently, the proposed GA-based model is compared with other optimization tools [6, 7]. Finally, the practicality of GA-based model in designing the 1 MW geothermal power plant is presented. In short, this paper contributes to a new application on the use of GA in designing a cost optimized the 1 MW geothermal system.

The rest of this paper is organized as follows. Mathematical formulations of shell-tube heat exchangers are presented in Section 2. The GA-based optimization for heat exchangers is explained in Section 3. The proposed model is validated in Section 4. A Case study is presented in Section 5. Finally, conclusions are presented in Section 6.

2. Mathematical model

In this section, some mathematical modeling pertaining to thermal, pressure and economic analysis of shell-tube heat exchanger are presented, as follows.

2.1 Thermal design of shell-tube heat exchanger

Energy balance equation for a shell-tube heat exchanger is expressed in Eq. (1) [33]:

$\begin{split}\displaystyle\dot{Q}&\displaystyle=\dot{m}_{h}C_{p,h}\left(T_{h,% in}-T_{h,out}\right)\\ &\displaystyle=\dot{m}_{c}C_{p,c}\left(T_{c,out}-T_{c,in}\right)\end{split}$ (1)

Heat exchanger surface area is computed using Eq. (2) [34]:

$A=\frac{\dot{Q}}{U\Delta T_{\textit{ML}}F}$ (2)

where $\dot{Q}$ is heat duty (W), $U$ is overall heat transfer coefficient (W/m ${}^{2}$ K), ${\Delta T}_{\textit{ML}}$ is mean logarithmic temperature difference ( ${{}^{\circ}}$ C), $F$ is temperature difference corrective factor.

The overall heat transfer coefficient is calculated by using Eq. (3) [35]:

$U=\frac{1}{{\displaystyle\frac{1}{h_{s}}}+R_{(f,s)}+{\displaystyle\frac{d_{o}}% {d_{i}}}\left(R_{(f,t)}+{\displaystyle\frac{1}{h_{t}}}\right)}$ (3)

with the assumption $d_{o}=1.25d_{i}$ [6, 7, 8, 12, 13]

Tube side heat transfer coefficient $h_{t}$ is computed by using Eqs (4)–(6) [35]:

$\displaystyle h_{t}=\frac{\lambda_{t}}{d_{i}}\left[3.657+\frac{0.0677(\textit{% Re}_{t}\textit{Pr}_{t}d_{i}/L)^{1.33}}{1+0.1\textit{Pr}_{t}(\textit{Re}_{t}d_{% i}/L)^{0.3}}\right]$ $\displaystyle\quad∼{}∼{}\mbox{ if }(\textit{Re}_{t}<2300)$ (4) $\displaystyle h_{t}=\frac{\lambda_{t}}{d_{i}}\left[\frac{f_{t}/8(\textit{Re}_{% t}-1000)\textit{Pr}_{t}}{1+12.7(f_{t}/8)^{0.5}(\textit{Pr}_{t}^{(2/3)}-1)}\right]$ $\displaystyle\quad∼{}\cdot\left[1+\left(\frac{d_{i}}{L}\right)^{0.67}\right]% \mbox{ if }(2300<\textit{Re}_{t}<10^{4})$ (5) $\displaystyle h_{t}=\frac{\lambda_{t}}{d_{i}}\cdot 0.027\textit{Re}_{t}^{0.8}% \textit{Pr}_{t}^{(1/3)}\left(\frac{\mu_{t}}{\mu_{t}w}\right)$ $\displaystyle\quad∼{}∼{}\mbox{ if }(\textit{Re}_{t}>10^{4})$ (6)

where $f_{t}$ is Darcy friction factor and determined by [8]:

$f_{t}=(1.82\log_{10}\textit{Re}_{t}-1.64)^{-2}$ (7)

$\textit{Pr}_{t}$ is Prandtl number for the tube side flow, which is given as follows [33]:

$\textit{Pr}_{t}=\frac{\mu_{t}C_{\textit{pt}}}{\lambda_{t}}$ (8)

Reynolds number of the flow in the tube side is calculated by using Eq. (9) [33]:

$\textit{Re}_{t}=\frac{\rho_{t}v_{t}d_{i}}{\mu_{t}}$ (9)

Flow velocity in the tube side is computed by using Eq. (10) [33, 35]:

$v_{t}=\frac{4(\dot{m}_{t})}{\pi d_{i}^{2}\rho_{t}}\cdot\frac{n}{N_{t}}$ (10)

where $n$ is the number of tube passes and $N_{t}$ is the number of tubes which is approximated based on Eq. (11) [7]:

$N_{t}=K_{1}\left(\frac{D_{s}}{d_{o}}\right)^{n_{1}}$ (11)

where $K_{1}$ and $n_{1}$ are coefficients taking values based on the flow arrangement and number of tube passes. Table 1 shows the values of $K_{1}$ and $n_{1}$ for triangular and square tube pitches.

Table 1

Value of $K_{1}$ and $n_{1}$ coefficient for $P_{t}=1.25d_{0}$ [8]

No. of passes	Triangle tube pitch		Square tube pitch
	$K_{1}$	$n_{1}$	$K_{1}$	$n_{1}$
1	0.319	2.142	0.215	2.207
2	0.249	2.207	0.156	2.291
4	0.175	2.285	0.158	2.263
6	0.0743	2.499	0.0402	2.617
8	0.0365	2.675	0.0331	2.643

Shell side heat transfer coefficient $h_{s}$ is determined by Eq. (12) [35]:

$h_{s}=0.36\frac{\lambda_{s}}{D_{e}}\textit{Re}_{s}^{0.55}\textit{Pr}_{s}^{(1/3% )}\left(\frac{\mu_{s}}{\mu_{\textit{sw}}}\right)^{0.14}$ (12)

where $D_{e}$ is shell hydraulic diameter which is defined as follows [7]:

$\displaystyle D_{e}=\frac{4(P_{t}^{2}-\pi d_{o}^{2}/4)}{\pi d_{o}}:\mbox{ % square pitch}$ (13) $\displaystyle D_{e}=\frac{4(0.43P_{t}^{2}-\pi d_{o}^{2}/8)}{0.5\pi d_{o}}:% \mbox{ triangular pitch}$ (14)

where $P_{t}$ is tube pitch which is assumed as [18]

$P_{t}=1.25d_{o}$ (15)

The Reynolds number for shell side flow is determined as follows [33]:

$\textit{Re}_{s}=\frac{\rho_{s}v_{s}D_{e}}{\mu_{s}}$ (16)

where shell side velocity is calculated as follows [34]:

$v_{s}=\frac{\dot{m}_{s}}{a_{s}\rho_{s}}$ (17)

The cross area of the fluid flow $a_{s}$ is determined by the following equation [8]:

$a_{s}=\frac{D_{s}BC_{l}}{P_{t}}$ (18)

where $C_{l}$ is shell side clearance and determined by [8]:

$C_{l}=P_{t}-d_{o}$ (19)

The Prandtl number for the shell side flow is determined as follows [33, 35]:

$\textit{Pr}_{s}=\frac{\mu_{s}C_{\textit{ps}}}{\lambda_{s}}$ (20)

Mean logarithmic temperature difference (LMTD) is determined by [33]

$\Delta T_{\textit{ML}}=\frac{(T_{h,\textit{in}}\!-\!T_{c,\textit{out}})(T_{h,% \textit{out}}-T_{c,\textit{in}})}{\ln((T_{h,\textit{in}}\!-\!T_{c,\textit{out}% })/(T_{h,\textit{out}}\!-\!T_{c,\textit{in}}))}$ (21)

The correction factor $F$ is determined based on the flow configuration and shell type (here the shell type is $E$ , according the TEMA standards), as follows [35]:

$\hskip-8.535827ptF=\sqrt{\frac{R^{2}+1}{R=1}\frac{\ln\left({\displaystyle\frac% {(1-P)}{(1-\textit{PR})}}\right)}{\ln\left({\displaystyle\frac{(2-\textit{PR}+% 1-\sqrt{R^{2}+1})}{(2-\textit{PR}+1+\sqrt{R^{2}+1})}}\right)}}$ (22)

where $R$ and $P$ are defined as follows [35]:

$\displaystyle R=\frac{T_{h,\textit{in}}-T_{h,\textit{out}}}{T_{c,\textit{out}}% -T_{c,\textit{in}}}$ (23) $\displaystyle P=\frac{T_{c,\textit{out}}-T_{c,\textit{in}}}{T_{h,\textit{in}}-% T_{c,\textit{in}}}$ (24)

Tube length $L$ is calculated by [35]:

$L=\frac{A}{\pi d_{o}N_{t}}$ (25)

2.2 Pressure drop in the shell-tube heat exchanger

Pressure drop in the tube side of the shell-tube heat exchanger is determined as the sum of distributed pressure drop along the tube lengths and local pressure losses in the elbows and in the inlet and outlet nozzles. It is expressed in Eq. (26) as given by [34].

$\hskip-8.535827pt\Delta P_{t}=\Delta P_{\textit{dis}}+\Delta P_{\textit{local}% }=\frac{\rho_{t}v_{t}^{2}}{2}\cdot\left(\frac{L}{d_{i}}f_{t}+p\right)\cdot n$ (26)

Different values of constant $p$ can be found by different authors. Kern [36] assumed $p=4$ , while Sinnott [37] assumed $p=2.5$ .

Shell side pressure drop can be determined by Eq. (27) as given by [18]

$\Delta P_{s}=\frac{\left(\rho_{s}v_{s}^{2}\right)}{2}\cdot f_{s}\cdot\left(% \frac{L}{B}\right)\cdot\left(\frac{D_{s}}{D_{e}}\right)$ (27)

where the friction factor $f_{s}$ is calculated as follows [8, 38]

$f_{s}=2b_{o}\textit{Re}_{s}^{-0.15}$ (28)

where $b_{o}=0.72$ [38] is valid for $\textit{Re}_{s}<$ 40000.

The required pumping power for the shell-tube heat exchanger is obtained as follows [12]:

$W_{\textit{pump}}=\frac{1}{\eta_{\textit{HE}}}\left(\frac{\dot{m}_{t}}{\rho_{t% }}\Delta P_{t}+\frac{\dot{m}_{s}}{\rho_{s}}\Delta P_{s}\right)$ (29)

2.3 Economic analysis of the shell-tube heat exchanger

The total cost of the shell-tube heat includes the investment cost and total discounted operating cost which is calculated as follows [13].

$C_{\textit{tot}}=C_{\textit{invest}}+C_{\textit{od}}$ (30)

The capital investment cost is a function of the heat exchanger surface according to Hall’s correction [39].

$C_{\textit{invest}}=a_{1}+a_{2}A^{a_{3}}$ (31)

where $a_{1}=8000$ , $a_{2}=259.2$ and $a_{3}=0.91$ for the heat exchanger made of stainless steel for both shells and tubes [39].

The total discounted operating cost related to the annual operating cost owning to the pumping power to overcome the friction flow in the heat exchanger can be calculated by the following equations [18]:

$\displaystyle C_{\textit{od}}=\Sigma_{j=1}^{\textit{yrs}}\frac{C_{o}}{(1+i)^{j}}$ (32) $\displaystyle C_{o}=W_{\textit{pump}}C_{e}H$ (33)

where $C_{e}$ is energy cost (€/kWh), $H$ is annual operating time (h/year), $i$ is the annual discount rate (%), $W_{\textit{pump}}$ is pumping power (kW) and computed by Eq. (30).

3. Methodology of the proposed ga-based model

In this section, the fitness function is described in Section 3.1 and the proposed GA-based model is explained in Section 3.2.

3.1 Fitness function

The purpose of this study is to obtain optimized design variables for designing a geothermal power system and subsequently achieve the minimization in total cost. The fitness function is defined by Eq. (34).

$\mbox{Minimization }f=C_{\textit{tot}}$ (34)

In this design, three main aspects were considered, i.e., thermal, pressure and economic, as discussed earlier in Section 2.1, Sections 2.2 and 2.3, respectively. To achieve an optimized design from the three aspects, this study focuses on four design variables, i.e., number of tube passes, tubes length, baffles spacing and tube outside diameter.

3.2 The proposed GA-based model

The term genetic algorithm (GA) was first proposed by Holland [40] which has widely applied to optimization and search problems in engineering design. It is inspired by investigating the mechanism of natural selection where stronger individuals would be the winners in a competitive environment.

Figure 1.

Flowchart of optimization design based on GA.

In order to achieve fitness function as expressed in Eq. (34), this study adopts GA. The methodology of the proposed GA-based model is simplified and illustrated in Fig. 1. It can be classified into three main stages, i.e., initialization, evolution and decision. In this first stage, a fitness function is defined based on the three aspects and the mathematical equations as presented in Eqs (1)–(34) are inserted in here. Subsequently, the GA-based model begins with the initialization of a random population. The population is an array of individuals, in this proposed model, the initial population size is set to 80 individuals. Each individual represents the candidate solutions pertaining to a problem where each of these solutions is analogous to a chromosome. Each chromosome is further associated with a set of variables or called genes. In this study, the chromosome refers to a set of a design solution for a shell-tube heat exchanger where gene refers ${N}_{t}$ , $L$ , $B$ and $d_{o}$ .

Subsequently, the algorithm evolves via three steps, i.e., selection, crossover and mutation. In the first step, every individual in the population is evaluated based on their fitness functions. The fittest solution or the solution that is better than others is selected. In the second step, the solutions from the previous step are mating one side to the other. This stage is to keep the good portion of a solution and subsequently combine with another good one. Hence, it is likely to produce an even better solution. The third stage introduces the random modification of these solutions. The purpose is to maintain diversity within the population and inhibit premature convergence. During this stage, individuals are selected and recombined through the crossover and mutation process, producing offspring which will comprise the next generation.

The second stage will iterate until a termination criterion is achieved. The termination criterion can be set a limit of generations or the computer clock time, to track the population’s diversity and stop when this falls below a preset threshold [41]. In this study, the number of generations is limited to 100. When a termination criterion is achieved, thus an optimized result is obtained. In this study, the minimized total cost is obtained and the best designs of ${N}_{t}$ , $L$ , $B$ and $d_{o}$ is suggested.

Table 2

Specifications of heat exchanger for validation case

Specifications	Shell side	Tube side
	Methanol	Sea water
Mass flow (kg/s)	27.8	68.9
T input ( ${}^{\circ}$ C)	95	25.0
T output ( ${}^{\circ}$ C)	40	40
Density (kg/m ${}^{3}$ )	750	995
$C_{p}$ (kJ/kgK)	2.84	4.20
$\mu$ (Pa.S)	0.00034	0.00080
$\mu_{w}$ (Pa.S)	0.00038	0.00052
$\lambda$ (W/mK)	0.19	0.59
$R_{\textit{fouling}}$ (m ${}^{2}$ K/W)	0.00033	0.00020

4. A simulation study with benchmark problem

The background of a benchmark problem is presented in Section 4.1. The results from our proposed model are presented and discussed in Section 4.2. A comparative study between the proposed model and other existing methods is also presented in Section 4.2.

4.1 Background

A simulation study using benchmark information from [6, 7] is adopted. The study investigates the design of heat exchanger for 4.34 MW. The specifications of the heat exchanger as presented Table 2 are the preliminary inputs of the proposed GA algorithm. For examples, the working fluids used in the shell and tube sides are methanol and brackish water, respectively. The mass flow rates of shell and tube sides are 27.8 kg/s and 68.9 kg/s, respectively. This heat exchanger has two tube side passages with triangle pitch pattern and one shell side passage.

For benchmarking purposes, this study uses the same settings as discussed in [6, 7], $\textit{yrs}=10$ , $j=$ 10%, $C_{e}=0.12$ €/kWh and annual amount of work hours $H=$ 7000 h/yr for all the experiments in this paper.

In order to validate the proposed model with and without TEMA standard, this study examines two types of variables, namely, discrete and continuous decision variables. The discrete decision variables are taken from TEMA standards, which are defined as follows:

b) a)
The number of tube passages adopts fours discrete values: 1, 2, 4 or 8 [12].
b)
The baffle spacing ( $B$ ) varies from the minimum baffle spacing of 0.0508 m to the maximum unsupported tube span of $29.5\times d_{o}^{0.75}$ where $d_{o}$ is in meters [12].
c)
The tube length ( $L$ ) adopts ten discrete values: 2.438 m, 3.048 m, 3.658 m, 4.877 m, 6.096 m, 7.32 m, 8.53 m, 9.75 m, 10.7 m and 11.58 m [42].
d)
The tube outer diameter ( $d_{o}$ ) adopts seven val-ues: 0.01588 m, 0.01905 m, 0.02223 m, 0.0254 m,0.03175 m, 0.0381 m and 0.0508 m [42].

In the case of imposing continuous decision variables, three variables are made continuously instead of rather than discrete values. The setting of the lower and upper bounds for the optimization variables are as follow: tube length ranging from 2.438 m to 11.58 m, tube outer diameter ranging between 0.01588 m and 0.0508 m, baffle spacing ranging from 0.05 m to 0.5 m.

Table 3
Model validation

Parameter Original Ref. [6] Ref. [7] Present work Present

design (without work

[37] TEMA) (TEMA)

$n$ 2 2 2 2 2

$D_{s}$ (m) 0.894 0.830 0.879 0.912 0.882

$L$ (m) 4.83 3.379 3.107 3.366 3.048

$B$ (m) 0.356 0.500 0.500 0.5 0.5

$d_{o}$ (m) 0.020 0.016 0.015 0.016 0.01588

$P_{t}$ (m) 0.025 0.020 0.01875 0.002 0.01985

$C_{l}$ (m) 0.005 0.004 0.00375 0.004 0.004

$N_{t}$ (m) 918 1567 3587 1503 1762

$v_{t}$ (m) 0.75 0.69 0.699 0.727 0.62

$\textit{Re}_{t}$ 14925 10,936 10,429 11483 9794

$\textit{Pr}_{t}$ 5.7 5.7 5.7 5.7 5.7

$h_{t}$ (W/m ${}^{2}$ K) 3812 3762 3864 4210 3355

$f_{t}$ 0.028 0.031 0.034 0.03 0.032

$\Delta P_{t}$ (Pa) 6251 4298 5122 5526 3860

$a_{s}$ (m ${}^{2}$ ) 0.032 0.0831 – 0.082 0.088

$D_{e}$ (m) 0.014 0.011 0.011 0.011 0.011

$v_{s}$ (m) 0.58 0.44 0.42 0.452 0.42

$\textit{Re}_{s}$ 18,381 11,075 9917 11253 10470

$\textit{Pr}_{s}$ 5.1 5.1 5.1 5.1 5.1

$h_{s}$ (W/m ${}^{2}$ K) 1573 1740 1740 1734 1693

$f_{s}$ 0.33 0.357 0.362 0.355 0.359

$\Delta P_{s}$ (Pa) 35,789 13,267 12,367 13309 11336

$U$ (W/m ${}^{2}$ K) 615 660 677 688 648

$A$ (m ${}^{2}$ ) 278.6 262.8 256.6 252.4 268

$C_{\textit{invest}}$ (€) 51,507 49,259 48,370 47,773 50,003

$C_{\textit{od}}$ (€) 12,973 5,818 5,995 5,652 4,437

$C_{\textit{tot}}$ (€) 64,480 55,077 54,366 53,425 54,440

4.2 Result and discussion

Parameter	Original	Ref. [6]	Ref. [7]	Present work	Present
	design			(without	work
	[37]			TEMA)	(TEMA)
$n$	2	2	2	2	2
$D_{s}$ (m)	0.894	0.830	0.879	0.912	0.882
$L$ (m)	4.83	3.379	3.107	3.366	3.048
$B$ (m)	0.356	0.500	0.500	0.5	0.5
$d_{o}$ (m)	0.020	0.016	0.015	0.016	0.01588
$P_{t}$ (m)	0.025	0.020	0.01875	0.002	0.01985
$C_{l}$ (m)	0.005	0.004	0.00375	0.004	0.004
$N_{t}$ (m)	918	1567	3587	1503	1762
$v_{t}$ (m)	0.75	0.69	0.699	0.727	0.62
$\textit{Re}_{t}$	14925	10,936	10,429	11483	9794
$\textit{Pr}_{t}$	5.7	5.7	5.7	5.7	5.7
$h_{t}$ (W/m ${}^{2}$ K)	3812	3762	3864	4210	3355
$f_{t}$	0.028	0.031	0.034	0.03	0.032
$\Delta P_{t}$ (Pa)	6251	4298	5122	5526	3860
$a_{s}$ (m ${}^{2}$ )	0.032	0.0831	–	0.082	0.088
$D_{e}$ (m)	0.014	0.011	0.011	0.011	0.011
$v_{s}$ (m)	0.58	0.44	0.42	0.452	0.42
$\textit{Re}_{s}$	18,381	11,075	9917	11253	10470
$\textit{Pr}_{s}$	5.1	5.1	5.1	5.1	5.1
$h_{s}$ (W/m ${}^{2}$ K)	1573	1740	1740	1734	1693
$f_{s}$	0.33	0.357	0.362	0.355	0.359
$\Delta P_{s}$ (Pa)	35,789	13,267	12,367	13309	11336
$U$ (W/m ${}^{2}$ K)	615	660	677	688	648
$A$ (m ${}^{2}$ )	278.6	262.8	256.6	252.4	268
$C_{\textit{invest}}$ (€)	51,507	49,259	48,370	47,773	50,003
$C_{\textit{od}}$ (€)	12,973	5,818	5,995	5,652	4,437
$C_{\textit{tot}}$ (€)	64,480	55,077	54,366	53,425	54,440

The proposed model has been applied to find the minimum total annual cost for the STHE. The results include optimal solutions with and without using the TEMA standards have been considered. The comparison results are shown in Table 3. Obviously, the results show a good agreement with the design solutions by the previous study [6, 7]. The results illustrate that energy saving up to 56.4% is possible by using the proposed configurations without adopting the TEMA standards, as compared with that of the original design.

Reductions of 2.9% and 5.7% in energy consumption can be achieved as compared with those from the optimal configurations by using the GA [6] and ICA [7], respectively. The capital investment can be reduced by 7.3%, 3.0%, and 1.2%, as compared with those from the designs in [6, 7, 37], respectively. Reduction of the total cost by 3.0% and 1.7%, respectively, can be achieved, as compared with those from [6, 7]. It is noted that the results obtained from [6, 7] and the proposed method without considering the TEMA are not practical though the total cost is slightly lower. For example, $d_{o}$ obtained based on the [6, 7] and the proposed method are 0.020 m 0.016 m 0.015 m and 0.016 m, respectively, while this parameter is suggested 0.1588 m by TEMA. On the other hand, values of $L$ are 4.83 m, 3.379 m, 3.107 m and 3.366 m, respectively, while the nearest available specification from TEMA is either 3.048 m or 3.658 m. Compare with the solution without using TEMA standard, the total cost based on TEMA is only 1.9% in different. The increase in price is still acceptable as the design solutions are suitable for industrial requirements.

In short, the results show good agreement with those from [6, 7] for the design with TEMA standard. Besides that, the implication of the proposed model in analyzing both cases i.e., with and without TEMA, is acceptable. Therefore, the proposed model with TEMA standard is further used in designing heat exchangers for 1 MW binary geothermal power plant in Section 5.

5. A case study

The background of a case study is introduced in Section 5.1. Subsequently, the results from the proposed model are further analyzed and discussed in two sections i.e., Sections 5.2 and 5.3, for preheater and evaporator, respectively. Finally, a remark is presented in Section 5.4.

5.1 Background

In this section, a case study pertaining to the geothermal power generation at Chingshui, Taiwan, is conducted. The focus of this study is on the optimal design of heat exchangers for the 1MW binary geothermal power system. Owing to the fact that Taiwanese manufacturer only produces materials based on the specification, thus the design of this case study directly adopts TEMA specification for practicality purpose. First of all, an initial survey pertaining to the resource at Chingshui is conducted. The survey shows that the brine inlet temperature is 100 ${}^{\circ}$ C with its specific heat capacity is 4.21 kJ/kg K. Upon discussion with domain experts, it is understood that the preheater and evaporator are among the two main important components in a geothermal power system. Hence, the design of the two components, i.e., preheater and evaporator, is studied. Both of these two components consist of shell and tube sides where shell side is the external surface that is filled with brine from geothermal well and tube side is the internal surface that is filled with working fluid. The working fluid, namely R134 is the most suitable selection according to its physical properties. From the available information, some studies have performed to design and select operating parameters for the overall binary geothermal power plant in general, for the preheater and evaporator in particular. The rest of the information as presented Table 4, which is the input of the proposed GA algorithm.

Table 4
Specifications of preheater and evaporator

Parameters	Preheater		Evaporator
	Shell side	Tube side	Shell side	Tube side
	Brine	R134a	Brine	R134a
Mass flow (kg/s)	66.8	48.53	66.8	48.53
$T$ input ( ${{}^{\circ}}$ C)	80	26.4	100	75
$T$ output ( ${{}^{\circ}}$ C)	66.7	75	80	75
Density (kg/m ${}^{3}$ )	974.85	1113.1	974.85	–
$C_{p}$ (kJ/kgK)	4.210	1.535	4.210	–
$\mu$ (Pa. s)	0.00038	0.00015	0.00038	–
$\mu_{w}$ (Pa. s)	0.00043	0.00012	0.67	–
$\lambda$ (W/mK)	0.67	0.072
$R_{\textit{fouling}}$ (m ${}^{2}$ K/W)	0.0002	0.0003	0.0002	0.003

Table 5

Optimal design for preheater and evaporator

Optimal variables
Parameters	Preheater		Evaporator
	Original design	Optimized design	Original design	Optimized design
$n$	2	8	2	8
$L$ (m)	4.877	2.438	6.096	3.658
$B$ (m)	0.71	0.5	0.6	0.5
$d_{o}$ (m)	0.01905	0.01588	0.0159	0.01588
Computational parameters
$D_{s}$ (m)	1.442	1.1024	1.41	1.14
$P_{t}$ (m)	0.023813	0.01985	0.019875	0.01985
$C_{l}$ (m)	0.004763	0.00397	0.004	0.004
$N_{t}$ (m)	3493	3078	5005	3374
$v_{t}$ (m)	0.137	0.894	0.154	0.912
$\textit{Re}_{t}$	15474.5	84272.59	32393	192215
$\textit{Pr}_{t}$	3.198	3.198	2.75	2.75
h ${}_{t}$ (W/m ${}^{2}$ K)	436	2028	483	2009
f ${}_{t}$	0.028	0.019	2.751	0.016
$\Delta P_{t}$ (Pa)	238.282	21621.35	320	23276
$a_{s}$ (m ${}^{2}$ )	0.205	0.11	0.17	0.114
$D_{e}$ (m)	0.014	0.011	0.011	0.011
$v_{s}$ (m)	0.335	0.622	0.404	0.601
$\textit{Re}_{s}$	11627.2	18001.8	11690	17395
$\textit{Pr}_{s}$	2.388	2.388	2.388	2.388
$h_{s}$ (W/m ${}^{2}$ K)	4028	6146	4932	6137
$f_{s}$	0.354	0.331	0.353	0.333
$\Delta P_{s}$ (Pa)	14118.5	29696.12	35726	43277
$U$ (W/m ${}^{2}$ K)	271	739	297	735
$A$ (m ${}^{2}$ )	1019.66	374.392	1522.1	615.7
$C_{\textit{invest}}$ (€)	149687	64933	212015	97530
$C_{\textit{od}}$ (€)	6309	19210	15895	26457
$C_{\textit{tot}}$ (€)	155996	84144	227910	123987

Figure 2.

The heat transfer coefficient of the optimal preheater and the original design.

Figure 3.

The total cost of the optimal preheater and the original design.

Figure 4.

The heat transfer of the optimal evaporator and the original design.

Figure 5.

The total cost of the optimal evaporator and the original design.

Figure 6.

The total cost reduction of the optimal preheater and the evaporator.

5.2 Analysis on preheater

The optimal design solutions for preheater and evaporator are tabulated in Table 5. It also presents the comparative results between the designs based on the conventional approach and the proposed approach. Column “Original design” indicates the conventional approach and column “Optimized design” indicates the proposed approach. As an example, it is observed that number of tube passes for preheater pertaining to the conventional design is 2 whereas the optimized design is 8. Though number of tube passes is increased, the total length of tube needed for the preheater has significantly reduced about 50% i.e., from 4.877 m reduced to 2.438 m. Therefore, a reduction of heat exchanger area is achieved owing to reduction of the heat exchanger length even the number of tube passes increased.

By using GA, the tube outer diameter decrease from 0.01905 m to 0.01588 m. The decrease of tube outer diameter increases the tube side heat transfer coefficient by 365.1% from 436 W/m ${}^{2}$ K to 2028 W/m ${}^{2}$ K with respect to the original design (see Fig. 2). Also, the shell side coefficient is observed to increase by 52.5%. The net result is 172.7% in overall heat transfer coefficient from 271 W/m ${}^{2}$ K to 739 W/m ${}^{2}$ K due to great increment of tube side heat transfer coefficient as compared to the original design.

The increment of overall heat transfer coefficient leads to 63.3% reduction in heat transfer area as compared to original design approach. Hence, the capital investment is reduced by 56.6%, i.e., from 149687 (€) to 64933 (€) as compared with that of the original design. This is mainly because of reduction in heat transfer area. A marked increase of pressure loss in both shell side and tube side is caused by the increment in number of tube passes, and an increase of flow velocity in both tube side and shell side. This pressure loss causes an increase of 204% from 6,309 (€) to 19,210 (€) in operating cost which is mainly caused by higher pumping power requirement for operating the heat exchanger. Thus, a reduction of 46.1% in the total cost is achieved by a cumulative effect of reduction in the capital investment and increment in operating cost, as compared with that of the original solutions. Figure 3 shows the comparison of the capital cost and the operating cost in original design solutions and optimal solutions.

5.3 Analysis on evaporator

The performance of GA in the case of evaporator optimization is showed in Table 5. Increasing the number of tube passes along with a reduction of heat exchanger length and number of tube, enabled to increase the heat transfer coefficient in both tube side and shell side. From Table 5, it is obviously seen that the tube side coefficient increase by 315.6%, i.e., from 483 W/m ${}^{2}$ K to 2009 W/m ${}^{2}$ K, as compared with that of the original design solutions. Also, the shell side coefficient increases by 24.4%. Hence, the overall heat transfer coefficient was increased by 147.3%. Figure 4 shows the comparison of the heat transfer coefficients between the original design solutions and the GA-based design.

A significant increase of flow velocity is observed in both tube side and shell side which leads to a great increase in pressure loss, as compared with that of the original design solutions. Thus, the operating cost increases by 66.4%, i.e., from 15,895 (€) to 26,457 (€), owing to a higher pumping power requirement for operating the heat exchanger. The capital investment reduces by 54%, i.e., from 212,015 (€) to 97,653 (€) due to a reduction of 59.5% in heat transfer area. Hence, the total cost reduces by 56.4% because of a cumulative effect of great reduction in the capital investment and increment of operating expenses compared to original design solutions. The comparison of the total cost is shown in Fig. 5.

5.4 Remark

By using the GA-based design conforming the TEMA standards, total capital investment cost for both the preheater and the evaporator reduces 55.1%, i.e., from 361,702 (€) to 162,463 (€), as compared with that of the original design (see Fig. 6). A remarked increment of total operating expenses is also observed (about 105.7%), i.e., from 22,204 (€) to 45,668 (€), because of a higher pumping power requirement for the configurations. Thus, a 52.8% reduction in total cost for both the preheater and the evaporator is achieved because of a cumulative effect of reduction in the investment cost and increment in the operating cost.

6. Conclusions

Heat exchanger plays a vital role in power plants. In this study, an optimal design and analysis pertaining to heat exchangers of an actual 1MW binary geothermal power plant are conducted. GA is employed to search for the optimal design of the shell-tube heat exchanger in economic aspect. First of all, the effectiveness and practicality of the proposed model are first validated using a benchmark problem from literature. The results have shown a good agreement with those from other studies [6, 7]. The design considers the total cost that includes capital investment and operating expenses for 10 years period. Subsequently, the proposed model is successfully used to optimize the preheater and the evaporator for designing 1MW the geothermal power plant. It shows that the area of heat transfer is reduced by 54.8% due to the increase of heat transfer coefficient. The decrease in the area leads to a significant reduction in total capital investment. It shows that the total capital investment is reduced by 55.1% as compared to the original design.

The results show that GA is suitable in identifying the optimized design for the heat exchangers. It provides faster and better efficiency than traditional method in handling multi-variable design problems which enable examining a number of alternative solutions of good quality and giving the designer more degrees of freedom in the final choice with respect to those from traditional methods.

In this study, the total cost is studied in term of Euro currency for benchmarking purposes. It is more appropriate to investigate the case study in New Taiwanese Dollar (NTD) as the geothermal power plant will be designed and made by local companies. Also, the annual discount rate should be revised according to the latest rate. For future works, it is interesting to observe the results between the optimized design with GA and the actual design of heat exchanger pertaining to the geothermal power plant. Through the observation, it can further improve the proposed GA-based method with considering some human reasoning and experience factors. It is also interesting to analyze the performance-effective design instead of cost-effective design only. Besides that, a user-friendly graphic user interface (GUI) optimization tool that assists engineers in design and development can also be developed.

Footnotes

Acknowledgments

This work is supported by the Taiwanese Ministry of Science and Technology (Grant No. MOST 103-3113-M-002-003).

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Optimal design for a shell-tube heat exchanger of a binary geothermal power plant from economic point of view

Abstract

Keywords

1. Introduction

2. Mathematical model

2.1 Thermal design of shell-tube heat exchanger

3.1 Fitness function

4.1 Background

5. A case study

5.1 Background

Table 4 Specifications of preheater and evaporator

5.3 Analysis on evaporator

5.4 Remark

6. Conclusions

Footnotes

Acknowledgments

References

Table 4
Specifications of preheater and evaporator