A Numerical Study for a Double Twisted Tube Heat Exchanger

Nowadays, there is an urgent need for highly intelligent and efficient heat exchangers. Various assumptions and techniques have been used to improve thermal performance and heat transfer in heat exchangers. The passive enhancement method of heat exchangers is commonly used in industries because it does not require more additional equipment or the consumed extra energy [1]. Some of the researchers have conducted studies on the twisted tube heat exchangers Any increased in the heat transfer that leads to the increasing pressure drop justifies the research in that area to evaluate the system's heat transfer and fluidic performance.

Mushatet and Hmood [2] numerically studied the effect of twist ratio on turbulent heat transfer in a twisted triangular tube heat exchangers. The results indicated that there was an increase in heat transfer in the twisted tube when compared with the regular tube. Dzyubenko et al. [3] made a twisted tube test to achieve the highest correlation between temperature and pressure drop at S / de = 6.2-34 and Re = 2000-40,000. Asmantas et al. [4] studied the heat transferred through a twisted oval tube. Their results showed improved heat transfer and flow resistance rates ranging between 20% -40% and 50% -80%, respectively. Tang et al. [5] studied heat transfer and pressure drop of twisted oval tube (TOT) and twisted tri-lobed tube (TTT) with the same twisted pitch p=200 mm. They conducted a numerical study on the effect of cross-section and twisted pitches of (TTT) on heat, and also examined the flow and heat transfer characteristics for different twisted direction. Many studies were conducted on the twisted square tube [6-9], the twisted spiral tube [10-12], and the twisted oval tube [13-16]. Many researchers have conducted extensive research on the twisted tapes, where they studied the effect of the twisted tapes inside the smooth circular tubes on the heat transfer and the pressure drop, Meyer et al. [17] made an experimental study on the characteristics of heat transfer and pressure drop in transitional regime of twisted tapes in smooth circular tubes, the experiments were conducted in circular tubes with an inner diameter of 19 mm and a length of 5.27 m and insertion of a twisted tape with ratios of 3,4,5. The experiments results showed that when the twisted ratio was varied and the heat flux was constant, the heat flux increased the time required to transfer from the laminar to the transitional flow. It was also found that the friction factors increase as the twist ratio decreases. In contrast, when the twist ratio and Reynolds number remain constant, an increase in the heat flux decreased the friction coefficient. While Sivakumar et al. [18] conducted experimental was carried for measuring heat transfer, friction factor and Reynolds number in a concentric tube containing a twist tape. The different twist ratio y = 2.52, 3.00 and 3.20 had been studied. The results showed that the twisted tape is accessible high heat transfer by friction factor increases. Whereas, Ponnada et al. [19], conducted an experimental study with a sophisticated technology using a heat pipe that works as a heat transfer device that works within Reynolds number 3000-16000. The heat transfer enhancement, friction factor and thermal performance of water in ordinary tube are compared with twisted tapes, perforated twisted tapes and alternate axis perforated twisted tapes in round tubes. Mushatet et al. [20] studied experimentally and numerically the effect of new tapered configuration of twisted tape in a heated tubes on a thermal and hydrodynamic characteristic. The experimental and numerical results revealed that the Nusselt number increased significantly by 75% and 100% respectively, and the friction factor increased by 220% and 226%, respectively. While the experimental and numerical thermal performance factor increases by 1.19 and 1.37, respectively. Mushatet and Youssif [21] studied numerically effect flow a nanofluid in a circular tube equipped with a double twisted tape. The results found that the heat transfer improves with the double twisted compared with the single twisted with increased volume concentration. Whereas, a counter clockwise twisting tape offered better thermal performance than clockwise twisting. In addition, the Nusselt number and the friction factor increased with the decrease in twist ratio decreases. Khudheyer and Farah [22] investigated numerically the impact of a double twisted tape in a circular tube in which a nanofluid flows. The Reynolds number ranged from 5000 to 35000, the volume concentration of 0.5% and 4%, and twisted ratio (y/w) of 2, 4, and 6. The results showed that the Nusselt number, the friction factor, and overall performance all increased with the decrease of the twisted ratio. Also, the heat transfer improved when using the double twisted tape and Al₂O₃/water nanofluid by 181%. While highest overall performance value of 2.25 was observed at a volume concentration of Al₂O₃/water is 4%, and the twisted ratio of (Tr=2).

Many researchers have made inserts on the tubes by adding twisted tapes to improve the heat transfer efficiency of the exchanger, as it is easy to assemble and disassemble these twisted tapes [23-30]. According to the aforementioned literature study, there is no study documented on twisted double tube heat exchanger. So this study focus on enhancement the performance of circular double tube heat exchanger by twisting the inner tube with different twist rations (Tr=5, 10 and 15) with a Reynolds number range of 5000-30000 under turbulent flow regimes.

The governing equations for three-dimensional turbulent flow and heat transfer fields include the continuity, momentum, and energy equations, realized by the Cartesian coordinate system and described by full Navier-Stockes. All of the mentioned equations are listed as follows [2]:

3.1 Assumptions of flow

The steady-state turbulent flow assumptions were taken.

The following assumptions:

1- Steady-state flow.

2- Turbulent flow.

3- Three-dimensional flow.

4- Incompressible flow.

5- Body force is ignored.

$\frac{\partial }{\partial x}\rho u+\frac{\partial }{\partial v}+\frac{\partial }{\partial z}\rho w=0$             (1)

Momentum equations:

$\begin{aligned} \rho\left(\frac{\partial \mathrm{u}^{2}}{\partial \mathrm{x}}+\frac{\partial \mathrm{uv}}{\partial \mathrm{y}}+\frac{\partial \mathrm{uw}}{\partial \mathrm{z}}\right) & \\=&-\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+\frac{\partial}{\partial \mathrm{x}}\left(2 \mu_{e f f} \frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)+\frac{\partial}{\partial \mathrm{y}}\left(\mu_{e f f} \frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right) \\ &+\frac{\partial}{\partial \mathrm{z}}\left(\mu_{e f f} \frac{\partial \mathrm{u}}{\partial \mathrm{z}}\right)+\frac{\partial}{\partial \mathrm{y}}\left(\mu_{e f f} \frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right) \\ &+\frac{\partial}{\partial \mathrm{z}}\left(\mu_{e f f} \frac{\partial \mathrm{w}}{\partial \mathrm{x}}\right) \end{aligned}$                (2)

Energy equations:

$\rho\left(\frac{\partial \mathrm{uv}}{\partial \mathrm{x}}+\frac{\partial \mathrm{v}^{2}}{\partial \mathrm{y}}+\frac{\partial \mathrm{v} \mathrm{w}}{\partial \mathrm{z}}\right)=-\frac{\partial \mathrm{p}}{\partial \mathrm{y}}+\frac{\partial}{\partial \mathrm{x}}\left(\mu_{e f f} \frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right)+\frac{\partial}{\partial \mathrm{y}}\left(2 \mu_{e f f} \frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)+\frac{\partial}{\partial \mathrm{z}}\left(\mu_{e f f} \frac{\partial \mathrm{v}}{\partial \mathrm{z}}\right)+\frac{\partial}{\partial \mathrm{x}}\left(\mu_{e f f} \frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)+\frac{\partial}{\partial \mathrm{z}}\left(\mu_{e f f} \frac{\partial \mathrm{w}}{\partial \mathrm{y}}\right)$                   (3)

$\frac{\partial \mathrm{uT}}{\partial \mathrm{x}}+\frac{\partial \mathrm{vT}}{\partial \mathrm{y}}+\frac{\partial \mathrm{wT}}{\partial \mathrm{z}}=\frac{\partial}{\partial \mathrm{x}}\left(\Gamma_{\text {eff }} \frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right)+\frac{\partial}{\partial \mathrm{y}}\left(\Gamma_{\text {eff }} \frac{\partial \mathrm{T}}{\partial \mathrm{y}}\right)+\frac{\partial}{\partial \mathrm{z}}\left(\Gamma_{\text {eff }} \frac{\partial \mathrm{T}}{\partial \mathrm{z}}\right)$               (4)

$\Gamma_{e f f}=\Gamma+\Gamma_{t}$                   (5)

$\mu_{e f f}=\mu+\mu_{t}$                   (6)

3.2 Turbulence model

The standard k-ɛ model is used to treat the influence of turbulence on the flow. Standard k-ɛ widely used in heat transfer flow due to their feasible accuracy for a broad range of turbulent flows [29]. It was initially suggested by Launder and Spalding. Transport equations in the standard k-ɛ model are written as follows [2]:

Turbulence kinetic energy (k) equation:

$\rho\left(\frac{\partial}{\partial x}(k u)+\frac{\partial}{\partial y}(k v)+\frac{\partial}{\partial z}(k w)\right)=\frac{\partial}{\partial x}\left(\frac{\mu_{z}}{\sigma_{k}} \frac{\partial k}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\mu_{e} \partial k}{\sigma_{k}} \frac{\partial k}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\mu_{e} \partial k}{\sigma_{k}} \frac{\partial k}{\partial z}\right)+G-\rho \varepsilon$               (7)

$\rho\left(\frac{\partial}{\partial \mathrm{x}}(\varepsilon \mathrm{u})+\frac{\partial}{\partial \mathrm{y}}(\varepsilon \mathrm{v})+\frac{\partial}{\partial \mathrm{z}}(\varepsilon \mathrm{w})\right)=\frac{\partial}{\partial \mathrm{x}}\left(\frac{\mu_{\mathrm{t}}}{\sigma_{\varepsilon}} \frac{\partial \varepsilon}{\partial \mathrm{x}}\right)+\frac{\partial}{\partial \mathrm{y}}\left(\frac{\mu_{\mathrm{t}}}{\sigma_{\varepsilon}} \frac{\partial \varepsilon}{\partial \mathrm{y}}\right)+\frac{\partial}{\partial \mathrm{z}}\left(\frac{\mu_{\mathrm{t}}}{\sigma_{\varepsilon}} \frac{\partial \varepsilon}{\partial \mathrm{z}}\right)+C_{1 \varepsilon} \rho \frac{\varepsilon}{\mathrm{k}} \mathrm{G}-\mathrm{C}_{2 \varepsilon} \rho \frac{\varepsilon^{2}}{\mathrm{k}}$                    (8)

where, G is referred to the generation term and is given

$\mathrm{G}=\mu_{\mathrm{t}}\left[2\left(\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)^{2}+2\left(\frac{\partial \mathrm{v}}{\partial \mathrm{y}}\right)^{2}+2\left(\frac{\partial \mathrm{w}}{\partial \mathrm{z}}\right)^{2}+\left(\frac{\partial \mathrm{v}}{\partial \mathrm{y}} \frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)^{2}+\left(\frac{\partial \mathrm{v}}{\partial \mathrm{z}} \frac{\partial \mathrm{w}}{\partial \mathrm{x}}\right)^{2}+\left(\frac{\partial \mathrm{v}}{\partial \mathrm{z}} \frac{\partial \mathrm{w}}{\partial \mathrm{y}}\right)^{2}\right]$                (9)

Also,

$\mathrm{k}=\frac{1}{2}\left(\overline{\mathrm{u}^{\prime 2}}+\overline{\mathrm{v}^{\prime 2}}+\overline{\mathrm{w}^{\prime 2}}\right)$                   (10)

$\varepsilon=\overline{\mathrm{e}_{1 j}^{\prime} \cdot \mathrm{e}_{1 j}^{\prime}}$               (11)

$\mu_{\mathrm{t}}=\rho c_{\mu} \frac{\mathbf{k}^{2}}{\varepsilon}$                    (12)

where, $c \mu=0.09, c 1 \varepsilon=1.44, c 2 \varepsilon=1.92, \sigma k=1, \sigma \varepsilon=1.3$.

3.3 Numerical component

The present model's governing equations are solved numerically. The Navier-Stokes equations in three dimensions are resolved by using the CFD code Fluent 18.2 predicted the flow behavior. Fluid flow parameters may be determined at a large number of places in a twisted tube using CFD modeling. The convergence standard for momentum and energy equations is defined as 1*10^-4.

3.4 Boundary conditions

The studied model was subjected to the boundary conditions as bellow:

1. at the inlets: mass-inlet and temperature-inlet is set as a boundary condition of inlets in two cases:

(a) mc is selected as (0.0827,0.1654,0.248,0.33,0.4135,0.4962) kg/s.

(b) mh is selected as (0.0984,0.1969,0.296,0.3938,0.4923,0.5908) kg/s.

(c) Tc is selected as (298) and Th is selected as (343).

2. At walls: no-slip condition, the outer wall of the outer pipe is insulated.

3. At the outlets: Relative gage pressure is assumed to be zero.

3.5 Data reduction

The logarithmic mean temperature difference (LMTD) is described as:

$\mathrm{LMTD}=\frac{\Delta \mathrm{T} 1-\Delta \mathrm{T} 2}{\ln \left(\frac{\Delta \mathrm{T} 1}{\Delta \mathrm{T} 2}\right)}$                 (13)

$\Delta T 1=(T h i-T c o)$                 (14)

$\Delta T 2=( Tho - Tci )$                 (15)

The heat exchanger effectiveness is: 

$\epsilon=\frac{\text { qact }}{q_{\max }}$                   (16)

$C h=m_{h} c p_{h}$

$C \mathrm{c}=m_{c} c p_{c}$

$qact=m \mathrm{Cc}(T c o-T c i)$                  (17)

$q m a x=(T h i-T c i)$                    (18)

where, C_min is the minimum value of Ch and Cc.

$\Delta P i=P h i-P h o$                (19)

$\Delta P o=P c i-P c o$               (20)

$\Delta P \mathrm{~T}=\Delta P_{\mathrm{o}}+\Delta P i$                  (21)

The overall performance index is given as 

$\eta=\frac{\epsilon}{\Lambda \mathbf{P T}}$                (22)

$R e=\frac{\rho \mathrm{Dh}, \mathrm{V}}{\mu}$                (23)

$D h, \mathrm{i}=d i$              (24)

${D h, \mathrm{o}}=D i-d o$                 (25)

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Figure 8. Temperature contours for double circular twisted tubes with twisted ratio 5

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Figure 9. Streamlines in the double circular twisted tube with elliptical cross-section and twist ratio of 5

The author would like to thank to my dear supervisor Prof. dr. Khudheyer S. Mushatet, Professor of heat transfer in the department of mechanical Engineering College of Engineering, University of Thi-Qar.