Investigation of Flow of the Disc-and-Doughnut Baffles and 40% Cut Segmental Baffles

2.1 Governing equations

The geometry is built by Multiphysics software’s drawing operations. Heat transfer theory was used to help investigate the heat transfer such as conduction, convection, thermal radiation and turbulent flow mechanism, also a combination of them or in collaboration with other physical properties [10]. This module can be applied using multiphysics coupling interface as temperature coupling between heat transfer and fluid flow interfaces is used in the computation.

The heat transfer was obtained from conservation law that complete the continuum mechanics theory; conservation of mass, conservation of linear momentum, and conservation of angular momentum [11, 12],

$\frac{\partial \rho}{\partial t} \nabla(\rho u)=0$    (1)

$\rho \frac{\partial u}{\partial t}+\rho(u \cdot \nabla) u=\nabla \cdot \sigma+F_{v}$    (2)

$\sigma^{T}=\sigma$    (3)

so, the localized heat balance equation in the spatial frame,

$\rho \frac{\partial E}{\partial t}+\rho u \cdot \nabla E+\nabla \cdot\left(q+q_{r}\right)=-(\sigma: D)+Q$    (4)

with $\sigma$ as the Cauchy stress tensor which are split into static and deviatoric parts,

$\sigma=-p I+\tau$    (5)

and the heat transfer in fluids interface are solved using,

$\rho C_{p}\left(\frac{\partial T}{\partial t}+u \cdot \nabla T\right)+\nabla \cdot\left(q+q_{r}\right)=\alpha_{p} T\left(\frac{\partial p}{\partial t}+u \cdot \nabla p\right)+\tau: \nabla u+Q$    (6)

where, ρ is the density (kg/m³), C_p is the specific heat capacity at constant pressure (J/(kg·K)), T is the absolute temperature (K), u is the velocity vector (m/s), q is the heat flux by conduction (W/m²), q_r is the heat flux by radiation (W/m²), α_p is the coefficient of thermal expansion (1/K),

$\alpha_{p}=-\frac{1}{\rho} \frac{\partial \rho}{\partial T}$    (7)

p is the pressure (Pa), τ is the viscous stress tensor (Pa), $\mathcal{Q}$ is the heat sources other than viscous dissipation (W/m³). In this case, the temperature does not change with time with a steady-state conditions.

Theoretically, the Nusselt number is suitable to calculate the heat transfer coefficient which is used to measure the energy of heat transfer in the system. There are two kinds of flow, inside the tube (hot fluid) and outside the tube – inside the shell (cold fluid). Inside the tube is calculated based on the result of Dittus-Boelter [13, 14] for internal flow, with n = 0.4 (shell) for heating and n=0.3 (tube) for cooling,

$N u_{t u b e}=0,023 R e_{D}^{4 / 5} \operatorname{Pr}^{n}$    (8)

While Nusselt number outside the tube or inside the shell is calculated by Churchill and Bernstein [15], this equation is used for large Reynolds number with value between 100 < Re_d < 10E7. The shell is composed of 14 tubes, with crossflow fluid.

$N u_{\text {shell }}=0.3+\frac{0.62 \cdot \operatorname{Re}^{1 / 2} \operatorname{Pr}^{1 / 3}}{\left[1+\left(\frac{0.4}{\operatorname{Pr}}\right)^{2 / 3} \quad \right]^{3 / 4}}\left[1+\left(\frac{R e}{282000}\right)^{5 / 8}\right]^{4 / 5}$    (9)

Reynolds number are investigated using the maximum velocity of the fluid inside the shell u_{max shell},

$R e_{\text {shell }}=\frac{\rho . u_{\text {max shell }} \quad . d_{\text {shell }}}{\mu_{\text {shell }}}$    (10)

with s_n as the vertical distance between the tube so,

$u_{\text {max.shell }}=u_{\text {shell }}\left(\frac{s_{n}}{s_{n}-d_{t u b e}}\right)$    (11)

The total area of heat transfer is calculated using the amount of the tube multiplied by the curve surface area of the tube, so,

$A_{t u b e}=N . \pi . d_{t u b e} . l$    (12)

where, N is the amount of the tubes (14 tubes) and l is the length of the tubes. The convection heat transfer coefficient of shell (h_shell) and the total heat transfer rate (q) may also be expressed theoretically,

$h_{\text {shell }}=\frac{N u_{\text {shell }} \cdot k_{f}}{d_{\text {shell }}}$    (13)

$q=h_{\text {shell }} A_{\text {tube }} \Delta T$    (14)

where, k_f is the conductivity of the fluid in the bulk temperature and $\Delta$T is the difference temperature of the both fluids. The effectiveness of the heat exchanger is a function of NTU (Number Thermal Unit) and the ratio of the minimum to the maximum value of heat capacity rate for hot and cold fluids (C_min/C_max).

$\varepsilon=\boldsymbol{f}\left(\boldsymbol{N T U}, \frac{\boldsymbol{C}_{\min }}{\boldsymbol{C}_{\max }}\right)$    (15)

2.2 Simulation setup

In this research, both water flow inside the tubes and outside the tubes (within the shell of the heat exchanger) are used. Both of the fluids have different inlet temperatures when entering the tubes and the shell. Hot water enters the tube side with velocity of 1, 3, 5, 10 m/s and temperature of 373.15 K. Cold water enters the shell side with velocity of 1, 3, 5, 10 m/s and temperature of 278.15 K. At both inlets, the value of turbulence length scale of 4.445 × 10^-4 m.

The radius of the shell and the tube are 0.055 m and 0.00635 m, respectively. The tubes are composed of 14 tubes with 0.583 m length, triangular 30^° rotated pitch. First, the baffles of disc-and-doughnut model, with radius of disc-and-doughnut of 0.055 m and 0.025 m, respectively. Second, the radius of 40% cut segmental baffles are 0.055 m. Both types, the distance between baffles of 0.116 m is evenly distributed along the shell. The total length of heat exchanger is 0.743 m. The inlet and outlet radius of both shell and tube are 0.025 m. Counter flow is used in the simulation. The outlet, both shell and tube are set on the normal flow with suppressed backflow.

The space dimension of the geometry was pictured using 3D axisymmetric with heat transfer and k - e turbulent flow of the physics problem. Stationary analyses were used for computation steps in this study. The material properties used were water and water, with Aluminum (Al) as the material of the baffles. The material properties of heat capacity at constant pressure, thermal conductivity, density, Young modulus, and ratio of specific heats are taken from an internal material database of both water and Aluminum as shown in Table 1.

Table 1. Material properties

Notes: The heat capacity at constant pressure and thermal conductivity for water is a polynomial function of the temperature.

Figure 1 shows the position of the baffles and inlet - outlet of the shell and the tube, respectively. The geometry mesh is also shown in the figure. The sequence type of mesh has to be created using user-controlled mesh with element size of extremely-coarse and finer meshes to make accurate solutions that are calculated using numerical techniques. The extremely-coarse element size meshes are distributed along the volume of the shell and stationary-head. The finer meshes are distributed along the tubes and even finer close to sharp edges at the cross section of the baffles with tubes and corners. The free tetrahedral complete mesh consists of 294788 domain elements, 30682 boundary elements, and 4205 edge elements, with minimum and maximum element size of 3.78 × 10^-4 m and 0.0035 m, respectively. Temperature, velocity, and pressure distribution are analyzed using COMSOL 5.2a. The calculation was solved using notebook with Intel^® Core^TM i7 Processors, 10^th Generation, 4.50 GHz, 16 GB of RAM. The total calculations time of 32 variations study case need 39 hours.

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Figure 1. Design of shell-and-tube disc-and-doughnut baffles (a) parts of shell-and-tube, (b) meshing symmetry

2.3 Profile

Heat transfer along the flow distribution can be shown and understood with the velocity and temperature profiles. Both of the profiles are drawn across the cross section along the length of shell-and-tube which are divided in 5 parts/lines and 5 points as shown in Figure 2, namely A 1–5, B 1–5, C 1-5, D 1-5, and E 1-5. These zones/positions are drawn to understand both the velocity and temperature profiles along the heat exchanger.

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Figure 2. Profile position of the points along of shell-and-tube and cross-section

The velocity of water (inlet) in shell side are 1, 3, 5, 10 m/s and tube side are 1, 3, 5, 10 m/s. To simplify, they are grouped and named with first and second numbers. The first numbers are the velocity of water entering the shell inlet, and second numbers are velocity of water entering the tube inlet. The legends used in Figure 9 and Figure 10 are Shell 1-Tube 1, Shell 1-Tube 3, Shell 1- Tube 5, Shell 1- Tube 10, Shell 3- Tube 1, Shell 3- Tube 3, Shell 3- Tube 5, Shell 3- Tube 10, Shell 5- Tube 1, Shell 5- Tube 3, Shell 5- Tube 5, Shell 5- Tube 10, Shell 10- Tube 1, Shell 10- Tube 3, Shell 10- Tube 5, and Shell 10- Tube 10.

In this research, numerical and theoretical simulation for heat exchanger with different type of baffles and variation of mass flow rates are performed to reveal the effect of different type of baffles on heat transfer and pressure drop characteristics. The baffle type of disc-and-doughnut baffles leads to turbulence of the fluid flow which causes increase in heat transfer characteristics and also have lower pressure drop than the 40% cut segmental baffles. Based on the theoretically, both type of disc-and-doughnut baffles and the 40% cut segmental baffles of heat exchanger that are investigated have the highest effectiveness at the lowest mass flow rate of the hot fluids (tube).