1 Introduction

Heat exchanger devices are used to transfer heat between two or more fluids which are at different temperatures and pressures, separated by a solid wall [1]. In heat exchanger apparatus, shell and tube heat exchanger (STHE) is playing key role (energy generation, oil refining industries, waste to heat recovery systems, etc.) due to robust construction of geometrical structure, ease of upgrade, and maintenance [2, 3]. To calculate the performance of STHE, baffle geometry and arrangement are essential components which are indicated in Fig. 1. Segmental baffle STHE is a commonly used one, and it will lead the fluid flow in tortuous and zigzag manner in the shell body [4,5,6]. These baffles are improving level of mixing to enhance the heat transfer of STHE which is given in Figs. 1 and 2.

Fig. 1
figure 1

Segmental baffle shell and tube heat exchanger

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Fig. 2
figure 2

Mesh geometry

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Ramananda Rao et al. [7] studied the shell and tube heat exchanger by choosing the parameters toward minimizing the cost of heat exchanger for any heat duty, author considered tube pitch (Pt) should be 1.25–1.5d0, minimum tube sheet thickness is 3.2 mm, length to diameter ratio should be 3 to 15 and baffle spacing should be 20% of Di to 100% of Di, it should not be less than 50 mm and finally concluded that these design parameters shown limited pressure drop and space and length constraints. Khalifeh Soltan et al. [8] listed out 20 and 100%Di. Saffar Avval et al. [9] presented an optimization program to calculate an optimum baffle spacing and the number of sealing strips for STHE, all most negligible effect found on the optimum pumping power and heat transfer, while the baffle spacing has a noticeable effect by author.

Gaddis et al. [10] presented a procedure to evaluate shell side pressure drop in segmental baffle shell and tube heat exchanger. Author varied the ratio of baffle spacing to shell inside diameter (S/Di) and the ratio of baffle cut to shell inside diameter (H/Di) within the range and concluded that \(0.2 \le \left( {\frac{S}{{D_{i} }}} \right) \le 1\) and \(0.15 \le \left( {\frac{H}{{D_{i} }}} \right) \le 0.4\). Iyer et al. [11] successfully established and designed optimal STHE. Kallannavar et al. [12] experimentally studied that STHE with different tube layout as 300, 450, and 600. Among these three tube layouts, 300 degree tube layout proved better heat transfer rate. Emad et al. [13] experimentally studied STHE with four different segmental baffle (conventional single segmental baffle, staggered single segmental baffle, flower segmental baffle, and hybrid segmental baffle) configurations to enhance the thermal, hydraulic, and thermodynamic performances. Hybrid segmental baffle configuration STHE is the best among the studied STHE. Gugulothu et al. [14] numerically studied different hydrodynamic characteristics and local parameters of a 3D geometry of shell and tube heat exchangers with segmental baffle.

The objective of the study is to investigate the heat transfer rate of different configured baffled spacing and provide the potential configuration through optimization study. For the investigation, the baffle spacing varied from 50 to 100 mm. The shell side flow rate varied from 0.18 to 0.31 kg/s, whereas the tube side flow rate varied from 0.11 to 0.18 kg/s. The study will put a remark to choose the best configuration of baffle spacing and flow rate considering both the variation of heat transfer rate and pressure drop.

2 Simulation and Modeling

2.1 Governing Equations

In the present investigation, the heat transfer and fluid flow phenomena on the shell side and tube side are studied in shell and tube heat exchanger with segmental baffle. The conservation equation, continuity, momentum, and energy are presented here.

Continuity equation:

$$\frac{{\partial \left( {\rho f_{v} } \right)}}{\partial t} + \frac{{\partial \left( {\rho f_{s} u_{j} } \right)}}{{\partial x_{j} }} = 0$$
(1)

Momentum equation:

$$\frac{{\partial \left( {\rho f_{v} u_{i} } \right)}}{\partial t} + \frac{{\partial \left( {\rho f_{s} u_{i} u_{j} } \right)}}{{\partial x_{j} }} = - f_{s} \frac{{\partial p_{i} }}{{\partial x_{i} }} + \frac{{\partial \left( {f_{s} \mu_{{{\text{eff}}}} \left( {\frac{{\partial u_{i} }}{{\partial x_{j} }} + \frac{{\partial u_{j} }}{{\partial x_{i} }}} \right)} \right)}}{{\partial x_{j} }} - 0.5f_{i} \rho \left| {u_{i} } \right|u_{i}$$
(2)

Energy equation:

$$\frac{{\partial \left( {\rho c_{p} f_{v} T} \right)}}{\partial t} + \frac{{\partial \left( {\rho c_{p} Tf_{s} u_{j} } \right)}}{{\partial x_{j} }} = \frac{{\partial \left( {f_{s} \lambda_{{{\text{eff}}}} \left( {\frac{\partial T}{{\partial x_{j} }}} \right)} \right)}}{{\partial x_{j} }} + \Omega K\left( {T_{t} - T} \right)$$
(3)

where u is velocity

2.2 Computational Domain

The STHE as shown in Fig. 1 is a single shell and tube pass with four tubes. The geometry of STHE is shell inner diameter (Di)=100 mm, length (Ls)=1150 mm, and tube bundle consists with ten tubes of 19 mm outer diameter in triangular arrangement with 25-mm tube pitch.

2.3 Boundary Conditions

The shell and tube heat exchanger consists of two loops carrying the coolant. The cold fluid flows in the shell with a temperature of 303.15 K. The hot fluid flows in four tubes at a temperature of 338.15 K. Several assumptions are made to simplify the geometry and study the numerical computational domain. Those are insulated shell walls, neglected fluid flow leakage, and fully developed fluid flow.

2.4 Mesh Selection and Sensitivity Analysis

STHX is a complex structure, so unstructured tetra-hydrate mesh was selected which is a proper grid system to mesh numerical models, i.e., shown in Fig. 2. Very fine mesh is done nearby tube wall region to obtain more accurate results. The finite volume method with SIMPLE pressure velocity coupling algorithm is used to solve the PDEs. PRESTO method with double precision solver was implemented to solve the convective terms in the governing equations [15,16,17].

2.5 Data Reduction

Tube outer diameter (d0) is 16 mm taken,

$${\text{Shell side Nusselt number}}\,{\text{Nu}}_{s} = 1.86{\text{Re}}_{s}^{0.33} \Pr_{s}^{0.33} \left( {\frac{{D_{i} }}{L}} \right)^{0.33}$$
(4)
$${\text{Shell side Reynolds number}}\,{\text{Re}}_{s} = \frac{{D_{e} V_{s} }}{{\vartheta_{s} }}$$
(5)
$${\text{Effective diameter (m)}}\,D_{e} = \frac{{P_{t}^{2} - \frac{{\pi d_{0}^{2} }}{4}}}{{\frac{{\pi d_{0} }}{4}}}$$
(6)
$${\text{According to TEMA standards }}P_{t} = 1.25d_{0} \,{\text{in mm}}$$
(7)
$${\text{Shell side Prandtl number }}Pr_s = \frac{{\mu_{s} C_{{p_{s} }} }}{{k_{s} }}$$
(8)
$${\text{Free flow area }}A_{s} = \frac{\pi }{4}\left( {D_{i}^{2} - N_{t} d_{0}^{2} } \right)$$
(9)
$${\text{Number of tubes }}N_{t} = K_{1} \left( {\frac{{D_{s} }}{{d_{0} }}} \right)^{{n_{1} }}$$
(10)

No. of passes

Triangular pitch

Square and rotated square pitch

K1

N1

K1

n1

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

$${\text{Shell outer diameter }}D_{s} = \frac{{D_{{{\text{otl}}}} }}{0.95} + \delta_{{{\text{clearence}}}}$$
(11)
$${\text{Tube bundle outer diameter }}D_{{{\text{otl}}}} = \left( {\frac{{N_{t} }}{{K_{1} }}} \right)^{{\frac{1}{{n_{1} }}}} d_{0}$$
(12)

According to Ravi et al., baffle space (BS) is 20–100%Di and it should not be less than 51 mm. In this research paper, geometry is created with different baffle spacing from 0.2 to 1Di and simulation is conducted.

$${\text{Heat exchanger surface area} ({\text {m}^{2}}) }A_{0} = \pi N_{t} d_{0} L$$
(13)
$${\text{Shell side Nusselt number Nu}}_{s} = 1.86{\text{Re}}_{s}^{0.33} Pr_{s}^{0.33} \left( {\frac{{D_{i} }}{L}} \right)^{0.33}$$
(14)
$${\text{Tube side Reynolds number }}{\text{Re}}_{t} = \frac{{d_{i} V_{t} }}{{\vartheta_{t} }}$$
(15)
$${\text{Tube side Prandtl number }}Pr_{t} = \frac{{\mu_{t} C_{{p_{t} }} }}{{k_{t} }}$$
(16)

3 Results and Discussions

3.1 Validation of Results

In order to validate the numerical model, the results of the present study were compared with the experimental study of Kamel et al. [18] following the same boundary conditions. The results of the comparison study are shown in Fig. 3. It is well recognized from the figure that the present study aligns precisely with a good agreement along with the study of Kamel et al. [18]. The maximum errors encountered during the simulation are found to be 14.6% at 0.18 kg/s flow rate, whereas the minimum error of 6.5% was encountered at 0.11 kg/s flow rate. Hence, following the results of the assessment, it could be said that the present numerical model could capture the thermo-hydraulic characteristics of the computational geometry accurately.

Fig. 3
figure 3

Comparison of the present study with experimental investigations

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3.2 Pressure Drop

The pressure distribution of heat exchangers can reflect the resistance characteristic and heat transfer performance. Figure 4 presents the pressure contour with superimposed fluid velocity. It is well recognized from the figure that along with the flow direction, the pressure drops linearly.

Fig. 4
figure 4

Shell side pressure of ms = 0.18 kg/s and mt = 0.11 to 0.18 kg/s

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Figure 5 shows the temperature distribution of all the studied six baffle space shell and tube heat exchangers with the segmental baffle. The temperature distribution is shallow behind the baffles, and local heat transfer is very poor. In Fig. 5, tube side temperature will gradually decrease from 338.15 to 322.618 K by increasing the shell side temperature until 314 K. It is noticed that the temperature is higher and gradually reduces near the shell wall center.

Fig. 5
figure 5

Temperature in x–y plane of ms = 0.18 kg/s and mt = 0.11 to 0.18 kg/s

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The velocity contours of computational flow domain are shown in Fig. 6 for the radial velocity at x=0.24 m distance. These contours are demonstrated in radial directions, and the authors noticed that due to the presence of the baffles, the vortex formations is observed which initiates the secondary flow in the flow regime. Such phenomenon of inducing secondary flow with the help of initiating and increasing turbulent intensity improves the heat transfer rate of the flow domain. Due to having lower space within the baffles, 50-mm baffle spacing induces higher secondary flow, which gradually decreases with the increase in baffle spacing and the least amount of secondary flow is evaluated for 100-mm baffle spacing. Consequently, it is recognized that 50-mm baffle spacing improves the heat transfer rate most, which gradually decreases with the increase in baffle spacing and least amount of improvement is observed for 100-mm baffle spacing. In addition, the upsurge in velocity gradient near the walls of the computational geometry is evident since the leakage zones in the heat exchanger decreases the velocity at shell center and increases at shell walls.

Fig. 6
figure 6

Velocity at x = 0.24 m of radial plane of ms = 0.18 kg/s and mt = 0.11 to 0.18 kg/s

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Figure 7 represents the average heat transfer and fluid flow characteristics across tube bundles in shell body at different baffle space from 50 to 100 mm. According to Fig. 7, heat transfer rate is increasing along with the increase in mass flow rate and baffle spacing. The fluctuation of the flow pattern increases with the decrease in baffle spacing, which eventually increases the turbulent intensity. Since the turbulent intensity plays as a predominant parameter in enhancement of heat transfer, lowering the baffle spacing induces higher heat transfer rate. As a result, the highest heat transfer rate was observed at 50-mm baffle spacing in Fig. 7, which progressively decreases with the increase in baffle spacing and achieved lowest value of heat transfer rate at 100-mm baffle spacing.

Fig. 7
figure 7figure 7

Average heat transfer rate of ms = 0.18 kg/s and mt = 0.11 kg/s

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Figure 8 indicates the pressure drop of all the studied shell and tube heat exchanger with different baffle spaces from 50 to 100 mm, i.e., 20–1%Di. From Fig. 8, it is identified that with the decrease in baffle spacing, the pressure drop increases. The maximum pressure drop was observed for 50-mm baffle spacing, which gradually decreases with the increase in baffle spacing, and lowest value of pressure drop was recognized for 100-mm baffle spacing.

Fig. 8
figure 8figure 8

Pressure drop of ms = 0.18 kg/s and mt = 0.11 kg/s

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4 Conclusion

In this research paper, work is done with different baffle space from 20 to 100%Di with segmental baffle shell and tube heat exchanger. In this work, 20%Di baffle space showed more pressure, high heat transfer, and 100%Di baffle space shown as less pressure and less heat transfer. Moreover, 70- and 80-mm baffle spacing in segmental baffle has shown the optimal solutions. The study will privilege the heat exchanger industries to implement the optimized configuration of baffle spacing, taking both heat transfer and pressure drop into consideration.