Modeling the Influence of Additives and Nanoparticles on Tubular Heat Exchanger Efficiency Utilizing Computational Fluid Dynamics

Heat transfer (HT) is one of the basic sciences that a large number of researchers are still interested in in most engineering fields due to its importance in various aspects of life and many engineering applications [1].

Heat is transferred in three ways: Conduction, convection, and radiation. These methods are characterized by specific patterns in which one or two heat transfer methods may participate together to contribute to the total HT. Understanding the exact mechanism of each HT method and studying the method of heat transfer through different media has been of interest to a large number of researchers due to its importance in improving the investment conditions for many thermal systems, whether through maximum heat transfer or comprehensive insulation, which reduces heat transfer to its minimum [2].

Heat transfer by the three methods is a huge issue that is affected by many factors, such as the medium that transfers heat, the type of materials that contribute to heat conduction, the physical state of the material, and many variables and factors that affect heat transfer. When studying heat transfer, there are two prevailing methods: the experimental method and the analytical method, and each of the two methods has its preference under specific conditions and constraints. The number of researchers working with the analytical approach has increased with the numerical modeling of heat transfer equations, especially after the great development in the world of computing and the emergence of new generations of processors capable of solving millions of equations with millions of unknowns in a relatively short time, which prompted a large number of researchers to adopt the analytical approach and simulate computational fluid dynamics (CFD) in their research [3].

CFD is a modern science that is witnessing extensive development in many engineering fields due to the development of computer technology and the ability of new generations of processors to work efficiently in solving a large number of equations [4].

The addition of nanoparticles to the working medium in thermal devices improves the thermal properties of these devices. Many materials can be used and added to water, such as graphene, Al₂O₃, Ti, TiO₂, Cu, Cuo, and other nanoparticles. As a result of the good thermal properties of these materials, they can raise the thermal properties of the nanofluid. Using graphene with water in particular increases the thermal conductivity (TC) of the new fluid because graphene has high TC. Therefore, adding graphene nanoparticles to water forms a fluid with high TC compared to water alone. It also improves the values of density, heat capacity, and other thermal properties associated with the new fluid [5].

The nanofluid produced by adding graphene to water in a specific volume ratio Has wide applicability in various engineering and heat-related fields, where it can be used in heat exchangers due to its major role in enhancing the heat transfer process. It is additionally suitable for applications involving heat pipes, sensors, refrigeration and air conditioning applications, lubrication, and other applications. Graphene-based nanofluid can be used in shell and tube heat exchangers (STE) to improve the efficiency of this type of exchanger [6].

Graphene nanofluids hold promise in high-performance, compact thermal systems, where maximizing heat transfer with minimal space and weight is critical. These systems include: Aerospace heat exchangers, microelectronics cooling systems, solar thermal collectors, high-efficiency heating, ventilation, and air conditioning (HVAC) systems, Automotive radiator applications.

The VF of nanoparticles Represents the ratio between the volume of nanoparticles and that of the entire fluid.

When studying a heat exchanger, we are dealing with the number of transfer units (NTU) rather than any non-dimensional number.

Bahmani et al. [7] studied heat transfer and turbulent flow of an aluminum oxide nanofluid in a double-pipe heat exchanger. The study showed that increasing the concentration of nanoparticles in the base fluid increased the heat transfer coefficient under both parallel and counterflow conditions. Rea et al. [8] demonstrated that using a nanofluid containing zirconium particles resulted in a 3% increase in the heat transfer coefficient, while the use of aluminum particles led to a 27% increase when heat transfer performance and pressure drop were assessed in a tube under laminar flow conditions. Das et al. [9] investigated the relationship between the temperature change of a nanofluid and the improvement of its thermal conductivity using Al₂O₃ and CuO. Goodarzi et al. [10] found that graphene exhibited the best thermal performance compared to other additives and was more economically preferable; the nanoparticles used in their study included Al₂O₃, graphene, Ti, TiO₂, Cu, and CuO. The effect of changing the type of nanoparticles added to water on the heat transfer coefficient and friction coefficient within a shell-and-tube exchanger was studied. Alawi et al. [11] experimentally investigated the energy efficiency of a flat-plate solar collector using graphene-based nanofluids (G-B-NFs), and the results showed an improvement in thermal properties and performance at constant heat flux. Ghozatloo et al. [12] conducted a thermodynamic evaluation (energy and exergy) of a shell-and-tube heat exchanger employing a graphene-based nanofluid, and demonstrated improved thermal performance under both laminar and turbulent flow conditions.

This research contributes to enhancing and improving the performance of heat exchangers by presenting a validated analytical-numerical methodology for solving them. While previous studies focused on numerical or experimental solutions, this methodology relied on both numerical and analytical solutions. The analytical and numerical results were compared to verify the validity of the numerical solution and provide a clear analytical methodology for studying shell-and-tube heat exchangers.

Also, most previous studies focused on aluminum, copper, and titanium oxides as nanomaterials added to the base fluid. However, this study used graphene as an additive, which is considered the least investigated nanomaterial when studying the thermal and hydraulic behavior of shell-and-tube heat exchangers. The study evaluates the effect of both nanoparticle VF and mass flow rate (MFR) on exchanger performance. This dual-criteria approach allows for a deeper understanding of operating conditions. The work utilizes a practical geometry (a shell-and-tube heat exchanger with realistic dimensions and boundary conditions), enhancing its relevance for industrial applications such as HVAC, process engineering, and power generation.

3.3 Energy equation

Conservation of energy in fluid flow systems is represented mathematically by the energy equation. It is based on the principles of thermodynamics and can be expressed as follows [3]:

$\rho\left(\frac{\partial E}{\partial t}+u \cdot \nabla E\right)=-\nabla \cdot(u(p+E))+\nabla \cdot(\kappa \nabla T)+\Phi$                            (3)

where,

$E$ is the internal energy.

$\kappa$ is the TC.

$T$ is the temperature.

$\Phi$ is the dissipation function.

3.4 k-omega turbulence model

It Relies on two partial differential equations describing the distribution of turbulence kinetic energy (k) and specific dissipation rate (ω).

Turbulent Kinetic Energy (k) Equation: Describes the energy contained in turbulence [2].

$\frac{\partial k}{\partial t}+u_j \frac{\partial k}{\partial x_j}=\frac{\partial}{\partial x_j}\left[\left(v+\frac{v_t}{\sigma_k}\right) \frac{\partial k}{\partial x_j}\right]+P_k-\beta^* k \omega$                         (4)

Specific Dissipation Rate (ω) Equation: Represents the rate of dissipation of turbulent kinetic energy per unit energy [4].

$\frac{\partial \omega}{\partial t}+u_j \frac{\partial \omega}{\partial x_j}=\frac{\partial}{\partial x_j}\left[\left(v+\frac{v_t}{\sigma_\omega}\right) \frac{\partial \omega}{\partial x_j}\right]+\alpha \frac{\omega}{k} P_k-\beta \omega^2$                      (5)

where,

$v$: Kinematic viscosity.

$v_t$: Turbulent viscosity.

$\sigma_k, \, \sigma_\omega$: Prandtl numbers for k and ω.

$P_k$: Turbulent production term.

$\alpha, \, \beta, \, \beta^*$: Model constants.

It is preferred to use it for flows where wall effects significantly impact turbulence. Widely used in fluid flow analysis through wind tunnels or around smooth surfaces where wall effects are prominent [9].

3.5 Analytical procedure

Let us first determine the rate of heat capacity of the water flowing in both the shell and tube calculated from: $C_s=$ $\dot{m}_s . C_w, C_t=\dot{m}_t . C_w$, where $C_s$ is the average heat capacity of water in the shell, $C_t$ is the average heat capacity of water in the tube and $\dot{m}_s \dot{m}_t$ is the mass flow rate of water in both the tube and the shell respectively [6].

Calculation of the maximum heat transfer rate is performed through the following formula:

$\dot{Q}_{\max }=C_{\min }\left(\mathrm{T}_{\max , i \mathrm{n}}-\mathrm{T}_{\min , i n}\right)=C_t\left(\mathrm{~T}_{t, i n}-\mathrm{T}_{s, i n}\right)$                       (6)

where, $T_{t, i n}$ and $T_{s, i n}$ are the water entry temperature in both the tube and the shell, respectively. Heat exchange surface area: $A=\pi D_{\text {inner}, t}. L$, where, $D_{\text {inner}, t}$ is the diameter of the inner tube and L is the length of the tube. The NTU is calculated as:

$N T U=\frac{U A}{C_{\min }}$                        (7)

where, UA it is the total heat transfer coefficients (HTC) [18].

To calculate the total HTC, we have a fluid in the inner tube, which is hot water, the cylindrical wall of the inner tube, and the cold fluid, which is also water in the shell.

Heat is transferred from the hot fluid inside the inner tube to its inner surface through convection. Subsequently, conduction occurs across the wall of the inner tube to the outer surface. Finally, heat is transferred by convection from the outer surface of the inner tube to the cold fluid within the shell. It is essential to determine the convection HTC for both the tube and the shell, which depend on the Reynolds, Prandtl, and Nusselt numbers, as well as to evaluate the thermal resistance across the thickness of the inner tube the HTC by convection of the fluid in the inner tube h_t calculated as [9]:

$h_t=\frac{N u_t \cdot K_t}{D_{\text {inner}, t}}$                         (8)

where, $K_t$ is the thermal conductivity coefficient of the fluid within the inner tube [19]. $N u_t$ is the Nusselt number (NN) of the fluid in the inner tube and $D_{\text {inner}, t}$ is the diameter of the inner tube [20]. The NN is related to the Prandtl and Reynolds number, as it relates to the type of flow within the tube, whether it is turbulent or laminar, and this is related to the RN, and therefore the RN must be calculated for the fluid inside the tube as follow [6]:

$R e_t=\frac{\rho V_{m, t} D_{\text {Inner}, t}}{\mu}$                         (9)

where, $V_{m,t}$ is the average velocity of flow in the inner tube and is calculated [2]:

$V_{m, t}=\frac{4 \cdot \dot{m}_t}{\rho \cdot \pi \cdot D_{\text {inner}, t}^2}$                        (10)

We note that the RN is $3.10^3<R e<5.10^6$, and therefore the flow is turbulent, then the NN is calculated from the Gnielinski relation [4]:

$N u=\frac{\left(\frac{f}{2}\right)(R e-1000) P r}{1+12.7\left(\frac{f}{2}\right)^{0.5}\left(p r^{2 / 3}-1\right)}$                         (11)

The coefficient of friction $f$ given by relation depending on the domain of the Reynolds number $10^4<R e_t<10^6$:

$f=\frac{1}{(1.58 \cdot \ln R e-3.28)^2}$                        (12)

Let us now recalculate in order to determine the convection coefficient of the cold fluid moving within the shell, the value of the average velocity of flow in the shell calculated as:

$V_{m, s}=\frac{4 . \dot{m}_s}{\rho . \pi .\left(D_{\text {inner}, s}^2-D_{\text {outer}, t}^2\right)}$                           (13)

where, $D_{\text {inner,} s}$ is the inner diameter of the shell, and $D_{\text {outer,} t}$ is the outer diameter of the tube. Thus, the RN of the fluid within the shell can be calculated from the relationship [7]:

$R e_s=\frac{\rho \cdot V_{m, s} \cdot D_{h y d}}{\mu}$                         (14)

where, $D_{h y d}$ is the hydraulic diameter of the annular space of the shell and is given by the relation [6]:

$\begin{gathered}D_{\text {hyd }}=4 \frac{A_{\text {flow }}}{\prod}=\frac{4 \cdot \frac{\pi}{4}\left(D_{\text {inner}, s}^2-D_{\text {outer}, t}^2\right)}{\pi\left(D_{\text {inner}, s}+D_{\text {outer}, t}\right)}=D_{\text {inner}, s}-D_{\text {outer}, t}\end{gathered}$                         (15)

where, $A_{\text {flow }}$ represents the cross-sectional area of the flow and $\prod$ represents the wetted perimeter.

Therefore, this flow is turbulent, and therefore the friction coefficient and NN is calculated from the same previous relationship based on hydraulic diameter.

The HTC due to convection in the fluid in the shell is then considered:

$h_s=\frac{K_s N u_s}{D_{h y d}}$                         (16)

Now the overall HTC is determined by the relationship [3]:

$\frac{1}{U A}=R=R_i+R_{\text {wall}}+R_o$                        (17)

$R_i$ and $R_o$ represent the convection thermal resistance of the internal and outer fluid respectively and given by:

$R=\frac{1}{h_s \cdot A}$                        (18)

$R_{wall}$ represents the thermal resistance of the inner wall of the tube and is given by:

$R_{\text {wall }}=\frac{\ln \left(\frac{D_{\text {outer}, t}}{D_{\text {inner}, t}}\right)}{2 \pi K_{\text {wall}} L}$                         (19)

$K_{wall}$ represents the TC of the inner wall material. Substitute in the relation for the total HTC [2]:

$R=\frac{1}{U A}=\frac{1}{\pi L}\left(\frac{1}{h_t D_{\text {inner}, t}}+\frac{\ln \left(\frac{D_{\text {outer}, t}}{D_{\text {inner}, t}}\right)}{2 K_{\text {wall}}}+\frac{1}{h_s D_{\text {outer}, t}}\right)$                        (20)

The heat capacity ratio is given by:

$c=\frac{c_{\text {min}}}{c_{\text {max}}}=\frac{c_t}{c_s}$                          (21)

The effectiveness of the STE with parallel and counter flow is calculated according to the relationship [18]:

$\varepsilon_{\text {parallel}}=\frac{1-e^{\left[-\frac{U A}{C_{\min}}\left(1+\frac{c_{\min}}{c_{\max}}\right)\right]}}{1+\frac{c_{\min}}{c_{\max}}}$                             (22)

$\varepsilon_{\text {counter }}=\frac{1-e^{\left[-\frac{U A}{C_{\min }}\left(1-\frac{c_{\min }}{c_{\max }}\right)\right]}}{1-\frac{c_{\min }}{c_{\max }} e^{\left[-\frac{U A}{c_{\min }}\left(1-\frac{c_{\min }}{c_{\max }}\right)\right]}}$                              (23)

For the VF of nanographene 0.005 by substituting into the previous relationships the effectiveness values of the heat exchanger are $\varepsilon_{\text {parallel }}=0.172$ and $\varepsilon_{\text {counter }}=0.174$.

3.6 Case study

In this paper, we will use a stainless-steel heat exchanger with parallel and counter flow, and the heat transfer process will take place between the hot fluid flowing in the inner tube and the cold fluid moving in the outer shell, as in Figure 1 and Table 1 showed these dimensions [20, 21]. The basic operating conditions for this exchanger are shown in Table 2. The physical properties of materials used are shown in the Table 3.

Table 1. Dimensions of heat exchanger

1.png

Figure 1. Geometry of shell and tube

Table 2. Operation condition

Table 3. Physical properties for materials

The analytical method will first be used to calculate the temperature at the outlet of the exchanger for the inner tube into which the hot water enters by using the NTU method of the shell-and-tube heat exchanger, and then compare the analytical result with the modeling results to validate the results. The properties of the nanofluid are calculated from the relationships [17]:

$\begin{gathered}\rho=(1-\varphi) \rho_w+\varphi \rho_{n p} \\ C_p=\frac{(1-\varphi) \rho_w C_{p w}+\varphi \rho_{n p} C_{p n p}}{(1-\varphi) \rho_w+\varphi \rho_{n p}} \\ K=\frac{k_w\left[k_{n p}+(n-1) k_w-(n-1) \varphi\left(k_w-k_{n p}\right)\right]}{k_{n p}+(n-1) k_w+\varphi\left(k_w-k_{n p}\right)} \\ \mu=\mu_w(1+2.5 \varphi)\end{gathered}$                          (24)

where, the suffix np stands for the properties of nanoparticles and the suffix w for water, but without the suffix, it is the properties of nanofluids. (Volume Fraction): It is the ratio of the volume occupied by nanoparticles to the total volume of the nanofluid. Graphene particles take the shape of flakes rather than the spherical shape. Therefore, when applying the Hamilton and Croser equation in the TC coefficient equation, a value of the particle shape coefficient n=1.14 must be used [22].