Effect of Heat Transfer on the Growing Bubble with the Nanoparticles/Water Nanofluids in Turbulent Flow

The field of nanofluids has novel properties of the fluid that make intensively useful in the field of heat transfer and its application including refrigerators, fuel cells, heat exchanger in the systems and conventional colloids [1-6]. It is appeared in the properties of thermal conductivity and convective heat/mass transfer. Measurably, the nanofluids have happened when the scale of measurement is smaller than 100nm functionalized nanoparticles, it is considered that a new class of material. It is known that the efficiency of heat transfer enhancement depends on the number of particles distributed, the type of the substance, the shape of particles, and so on. Bubble dynamics and nanofluids are planned to be used in aircraft, engines, micro-reactors [5, 6]. Prior to the introduction of nanofluids, it was expected that heat transfer could be enhanced by dispersing micron-sized particles. But the fluid of micron-sized particles caused problems due to sedimentation and clogging.

A lot of research has undergone for the last years in the field of boiling systems of nanofluids. However, for most potential applications of nanofluids the situation of interest is flow boiling including bubble dynamics parameters, such as bubble departure diameter, frequency and void fraction during nanofluid flow boiling, that may provide valuable insight in the mechanisms by which the nanoparticles affect the heat transfer [2].

Recently, it has become increasingly necessary to study various methods for increasing heat transfer especially in the field of bubble dynamics and their applications like peristaltic motion [7] and nuclear power system [5], resulting from the transfer of nanoparticles in common liquids called nanofluids. Measurements of nanoparticles are used in nanofluids range from 1:100 nm. These nanoparticles are metal particles such as Cu, Ag or other metal oxides. Typical fluids used in heat transfer usually have a low coefficient of heat transfer, but due to their high coefficient of conductivity, the use of nanoparticles can increase the coefficient of thermal conductivity of mixing fluids. The use of nanofluids is also a suitable approach to improve heat transfer [6, 8-10]. The distribution of temperature around the growth of a vapour bubble is an important process in nanoparticles/water (for example, Al₂O₃/H₂O, Fe₃O₄/H₂O and CuO//H₂O) nanofluids between a two-phase turbulent flow. In nanoparticles nanofluid, the growth process is influenced by heat transfer and pressure change between the laminar or turbulent flow of two phases. The problem of heat transfer in a growing bubble with initial conditions in the laminar flow is introduced and discussed in Ref. [11].

The bubble dynamics in nanofluids are studied experimentally and introduced by some efforts of researchers [5, 12-15]. Park et al. [12] attempted to formulate the problem of bubble dynamics in nanofluids and find experimentally the behaviours of boiling bubbles. Moreover single bubble is studied in Al₂O₃/H₂, Fe₃O₄/H₂O and CuO//H₂O nanofluids considering the impact of heating surface. In another side, some works (as in Refs. [16, 17] showed the model of the bubble dynamics in viscoelastic fluids that took Oldroyd constitutive equation on works [16, 17], which allows for strain relaxation of the fluid and is more suitable to greater deformations. The evaluation of radius under influence of shear stress is theoretically studied in ref. [18, 19]. Recently, Abu-Nab et al. [20] investigated the vapour bubble dynamic in power-law Al₂O₃/H₂ nanofluid, affecting by variable surface tension depending on the method of Plesset and Zwick [21] and Mohammadein et al. [19]. Furthermore, the effects of surface tension behavior with the initial bubble radius, time, and initial rate of bubble radius are investigated within Al₂O₃/H₂ nanofluid. It is found that the growth of the vapour bubble radius is affected by the concentration rate of nanoparticles and other parameters. However, the influence of heat transfer on growth process of bubbles in nanoparticles/water nanofluids between two-phase turbulent flow has not been studied analytically before. Consequently, we will take that into account.

The main aim of the current work is to study the effect of heat transfer on a growing bubble in Al₂O₃/H₂, Fe₃O₄/H₂O and CuO//H₂O nanofluids between two-phase turbulent flow and determine the bubble radius under the influence of thermo-physical parameters of superheating liquid, nanoparticles volume fraction and density in nanoparticles/water nanofluids. Besides, we illustrate the temperature distribution in the regime of bubble growth in turbulent flow, during nanoparticles/water nanofluids.

The introduction presented in section one. The theoretical study, mathematical model and solution of the model are introduced in section two and three. Results and discussions, and conclusion are addressed in section four and five respectively.

This section is to present the solution of theoretical model of the temperature distribution surrounding the growing bubbles in some different types of nanoparticles/water nanofluids, based on the reduced system of Eqns. (1-6) and the method of the similarity parameter between two finite boundaries, we will use the following as

$\mathrm{T}(\mathrm{r}, \mathrm{t})=\mathrm{T}(\mathrm{s}), \mathrm{r}=\frac{1}{\beta} \mathrm{s} \mathrm{f}(\mathrm{t})$ and $\frac{\partial \mathrm{s}}{\partial \mathrm{t}}=0$          (7)

At the boundary of bubble, we suppose r=R then

s=β             (8)

And r=f(t).

The temperature gradient in the mixture of nanofluids at boundary of bubble takes

$4 \pi \mathrm{R}^{2} \mathrm{k}_{\mathrm{nf}}\left(\frac{\partial \mathrm{T}}{\partial \mathrm{r}}\right)_{\mathrm{r}=\mathrm{R}}=\frac{\mathrm{d}}{\mathrm{dt}}\left[\frac{4 \pi \mathrm{R}^{3}}{3}\left(\mathrm{~L}+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right) \rho_{\mathrm{v}}\right]$          (9)

Combining (3) and (9), the temperature gradient $\left(\frac{\partial \mathrm{T}}{\partial \mathrm{r}}\right)_{\mathrm{r}=\mathrm{R}}$ becomes

$\left(\frac{\partial \mathrm{T}}{\partial \mathrm{r}}\right)_{\mathrm{r}=\mathrm{R}}=\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{v}}}{\mathrm{k}_{\mathrm{nf}}} \dot{\mathrm{R}}\left(\mathrm{L}+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right)$        (10)

On the basis of the above assumptions and applying the above assumptions and separating of variables methods on

Eq. (4), then the energy Eq. (4) converts to two equations, the obtained equation can be divided to two differential equations as 

$\dot{\mathrm{f}}(\mathrm{t}) \mathrm{f}(\mathrm{t})=\mathcal{A}$            (11)

And

$\frac{\mathrm{d}}{\mathrm{ds}} \ln \left(\frac{\mathrm{dT}}{\mathrm{ds}}\right)=\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left[-\frac{\mathrm{s}}{\beta^{2}}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \beta}{\mathrm{s}^{2}}\right]-\frac{2}{\mathrm{~s}}$          (12)

We integrate Eq. (12), we get

$\frac{\mathrm{dT}}{\mathrm{ds}}=\frac{\mathcal{A}}{\mathrm{s}^{2}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{\mathrm{s}^{2}}{2 \beta^{2}}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \beta}{\mathrm{s}}\right)\right]}$        (13)

The boundary condition (5) can be converted to

$\left(\frac{\mathrm{dT}}{\mathrm{dr}}\right)_{r=\mathrm{R}}=\frac{\mathcal{A} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{v}}\left(\mathrm{L}+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right)}{\mathrm{k}_{\mathrm{n} \mathrm{R}} \mathrm{R}}$      (14)

We obtain $\frac{\mathrm{dT}}{\mathrm{dr}}=\frac{\mathrm{s}}{\mathrm{r}} \frac{\partial \mathrm{T}}{\partial \mathrm{s}}$, thus

$\left(\frac{\mathrm{dT}}{\mathrm{ds}}\right)_{\mathrm{r}=\beta}=\left(\frac{\mathrm{r}}{\mathrm{s}} \frac{\mathrm{dT}}{\mathrm{dr}}\right)_{\mathrm{r}=\mathrm{R}, \mathrm{s}=\beta}=\frac{\mathcal{A} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{v}}\left(\mathrm{L}+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right)}{\mathrm{k}_{\mathrm{nf}} \beta}$           (15)

To find the constant $\mathcal{A}_{1}$, using (15) into (13), then

$\begin{aligned} \mathcal{A}_{1}=\frac{1}{\mathrm{k}_{\mathrm{nf}}} \mathcal{A} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{v}}(\mathrm{L}+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}\right.\left.\left.-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right) \beta \mathrm{e}\left[\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} c_{0}^{2}}\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right] \end{aligned}$ 

Using the value of constant $\mathcal{A}_{1}$ into Eq. (13), we get

$\frac{\mathrm{dT}}{\mathrm{ds}}=\frac{\mathcal{A}_{2} \mathcal{A} \beta}{\mathrm{s}^{2}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{\mathrm{s}^{2}}{2 \beta^{2}}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \beta}{\mathrm{s}}\right)+\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right]}$          (16)

where, $\mathcal{A}_{2}=\frac{1}{\mathrm{k}_{\mathrm{nf}}} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{v}}\left(\mathrm{L}+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right)$.

From Eq. (15), Eq. (16) leads to

$\frac{\partial \mathrm{T}}{\partial \mathrm{r}}=\frac{\mathrm{s}}{\mathrm{r}} \frac{\mathrm{dT}}{\mathrm{ds}}$

$=\frac{\mathcal{A}_{2} \mathcal{A} \beta}{\mathrm{r} \mathrm{s}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{\mathrm{s}^{2}}{2 \beta^{2}}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \beta}{\mathrm{s}}\right)+\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right]}$          (17)

Using the assumptions in (7,8), Eq. (17) becomes

$\frac{\partial \mathrm{T}}{\partial \mathrm{r}}=\frac{\mathcal{f}_{2} \mathcal{A} \mathrm{R}}{\mathrm{r}^{2}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{\mathrm{r}^{2}}{2 \mathrm{R}^{2}}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \mathrm{R}}{\mathrm{r}}\right)+\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right]}$             (18)

Eq. (18) can be integrated from any instant $\mathrm{t}$ to $\mathrm{t}_{\mathrm{m}}$ at $\mathrm{r}$ reaches to maximum radius R_m also T(r, t)=T₀, that means

$\mathrm{T}(\mathrm{r}, \mathrm{t})=\mathrm{T}_{0}-\mathcal{A}_{2} \mathcal{A}$

$\mathrm{R} \int_{\mathrm{r}}^{\mathrm{R}_{\mathrm{m}}} \frac{1}{\mathrm{z}^{2}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}} \cdot\left(\frac{\mathrm{z}^{2}}{2 \mathrm{R}^{2}}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \mathrm{R}}{\mathrm{z}}\right)+\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}} \cdot\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right]} \mathrm{dz}$           (19)

In the wall of bubble, we suppose r=R(t_i), Eq. (19) becomes

$\mathrm{T}\left(\mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right), \mathrm{t}_{\mathrm{i}}\right)=\mathrm{T}_{0}-\mathcal{A}_{2} \mathcal{A} \cdot \mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right)$

$\int_{\mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right)}^{\mathrm{R}_{\mathrm{m}}} \frac{1}{\mathrm{z}^{2}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} c_{0}^{2}} \cdot\left(\frac{\mathrm{z}^{2}}{2 \mathrm{R}^{2}\left(\mathrm{t}_{\mathrm{i}}\right)}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right)}{\mathrm{z}}\right)+\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}} \cdot\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right]} \mathrm{dz}$         (20)

Supposing that y=z/R(t_i) into Eq. (20), we get the temperature distribution of a growing bubble in nanoparticles nanofluids between two-phase flow in the form

$\mathrm{T}\left(\mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right), \mathrm{t}_{\mathrm{i}}\right)=\mathrm{T}_{0}-\mathcal{A}_{2} \mathcal{A} \int_{1}^{\frac{\mathrm{R}_{\mathrm{m}}}{\mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right)}} \frac{1}{\mathrm{y}^{2}} \mathrm{e}^{\left[\frac{-\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} c_{0}^{2}}\left(\frac{\mathrm{y}^{2}}{2}+\frac{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}}{\mathrm{y}}\right)+\frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\left(\frac{1}{2}+\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\right]} \mathrm{dy}$       (21)

By integrating the R. H. S of above equation, Eq. (21) becomes

$\mathrm{T}\left(\mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right), \mathrm{t}_{\mathrm{i}}\right)=\mathrm{T}_{0}-\mathcal{A}_{2}\left(\frac{\pi}{6} \mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2} \mathcal{A}\right)^{\frac{1}{2}} \mathrm{e}^{\frac{1}{6}\frac{A\left(1-\varepsilon_{n f} C_{k}\right)^{2}}{a_{\operatorname{Tur}} c_{0}^{2}}}$

$\left(\operatorname{erf}\left[\left(\frac{3}{2} \frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\right)^{\frac{1}{2}}\left(1-\frac{\mathrm{R}\left(\mathrm{t}_{\mathrm{i}}\right)}{\mathrm{R}_{\mathrm{m}}}\right)\right.\right. \left.+\left(1-\varepsilon_{\mathrm{nf}} C_{\mathrm{k}}\right)\left(\frac{1}{6} \frac{\mathrm{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\right)^{\frac{1}{2}}\right] \left.-\operatorname{erf}\left[\left(1-\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\right)\left(\frac{1}{6} \frac{\mathrm{c}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\right)^{\frac{1}{2}}\right]\right)$        (22)

At $\mathrm{t}=\mathrm{t}_{0}, \mathrm{R}\left(\mathrm{t}_{0}\right)=\mathrm{R}_{0} \Rightarrow \operatorname{erf}(0)=0$ and $\mathrm{T}\left(\mathrm{R}\left(\mathrm{t}_{0}\right), \mathrm{t}_{0}\right)=\mathrm{T}_{\mathrm{s}},$ Eq. (22) converts to

$\Delta \Omega=\mathrm{T}_{0}-\mathrm{T}_{\mathrm{s}}=\mathcal{A}_{2}\left(\frac{\pi}{6} \mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2} \mathcal{A}\right)^{\frac{1}{2}} \operatorname{erf}\left[\left(\frac{3}{2} \frac{\mathcal{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\right)^{\frac{1}{2}}\left(1-\phi_{0}^{1 / 3}\right)\right]$          (23)

Here $\phi_{0}=\left(R_{0} / R_{m}\right)^{3}$, is a void fraction. T_s is the saturation temperature.

Using the value of $\mathcal{A}_{2}$ into Eq. (23), we get

$\Delta f=\frac{1}{\mathrm{k}_{\mathrm{nf}}} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{V}}(\mathrm{L}\left.+\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right) \Delta \Omega\right)\left(\frac{\pi}{6} \mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2} \mathcal{A}\right)^{\frac{1}{2}} \operatorname{erf}\left[\begin{array}{l}\left(\frac{3}{2} \frac{\mathrm{A}}{\mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2}}\right)^{\frac{1}{2}} \\ \left(1-\phi_{0}^{1 / 3}\right)\end{array}\right]$          (24)

We note that, $\operatorname{erf}(\mathrm{x})=1-\frac{1}{\sqrt{\pi} \mathrm{x}} \mathrm{e}^{-\mathrm{x}^{2}}$ at any $\mathrm{x} \gg 0,$ also, for

any $\mathrm{a}_{\mathrm{Tur}} \ll 1,$ and for any times, we obtain $0<\phi_{0}^{1 / 3}<1$. Then Eq. (24) reduces

$\frac{\left(J_{\mathrm{a}}\right)_{\mathrm{nf}}\left(\mathrm{a}_{\mathrm{Tur}}\right)_{\mathrm{nf}}}{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\left(1+\frac{\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right)}{\mathrm{L}} \Delta \Omega\right.} =\left(\frac{\pi}{6} \mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2} \mathcal{A}\right)^{\frac{1}{2}}-\frac{1}{3\left(1-\phi_{0}^{\frac{1}{3}}\right)} \mathrm{e}^{\left[\left(\frac{-3}{2 \mathrm{a}_{\mathrm{Tur}} c_{0}^{2}}\right)^{\frac{1}{2}}\left(1-\phi_{0}^{1 / 3}\right)\right]}$            (25)

Here $\left(J_{a}\right)_{n f}=\frac{\rho_{n f}\left(a_{\text {Tur }}\right)_{\text {nf }}}{\rho_{\mathrm{v}} \mathrm{L}} \Delta \Omega$, is the Jacob number in nanofluids.

For a_Tur, we get

$\frac{\Delta \Omega \mathrm{k}_{\mathrm{nf}}}{\varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}} \rho_{\mathrm{v}}\left(1+\frac{\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right)}{\mathrm{L}} \Delta \Omega\right)}=\left(\frac{\pi}{6} \mathrm{a}_{\mathrm{Tur}} \mathrm{c}_{0}^{2} \mathcal{A}\right)^{\frac{1}{2}}$  

then $\mathcal{A}$ can be written as

$\mathcal{A}=\frac{6 \mathrm{a}_{\mathrm{Tur}}}{\pi}\left(\frac{\left(\mathrm{J}_{\mathrm{a}}\right)_{\mathrm{nf}}}{c_{0} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\left(1+\frac{\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right)}{\mathrm{L}} \Delta \Omega\right)}\right)^{2}$             (26)

Apply the boundary conditions at t=t₀, R=R₀ in solving Eq. (12a) to get the bubble radius in nanofluids, then we get

$\mathrm{R}=\left(2 \mathcal{A}\left(\mathrm{t}-\mathrm{t}_{0}\right)+\mathrm{R}_{0}^{2}\right)^{\frac{1}{2}}$           (27)

Combining (26, 27), we get the bubble radius in nanofluids between two turbulent flow in a final form

$\mathrm{R}=2\left(\frac{3 \mathrm{a}_{\mathrm{Tur}}}{\pi}\left(\frac{\left(\mathrm{J}_{\mathrm{a}}\right)_{\mathrm{nf}}}{\mathrm{c}_{0} \varepsilon_{\mathrm{nf}} \mathrm{C}_{\mathrm{k}}\left(1+\frac{\left(\left(\mathrm{C}_{\mathrm{P}}\right)_{\mathrm{nf}}-\mathrm{C}_{\mathrm{Pv}}\right)}{\mathrm{L}} \Delta \Omega\right)}\right)^{2}\left(\mathrm{t}-\mathrm{t}_{0}\right)\right. \left.+\frac{1}{4} R_{0}^{2}\right)^{\frac{1}{2}}$               (28)

The formula (28) denotes the behaviour of vapour bubble growth between two-phase turbulent flow under conditions in (7), that affected by nanoparticles nanofluids.

We present the results of the influence of heat transfer on bubble dynamics with some different types of nanofluids like $\mathrm{Fe}_{3} \mathrm{O}_{4} / \mathrm{H}_{2} \mathrm{O}, \mathrm{CuO} / \mathrm{H}_{2} \mathrm{O}, \mathrm{Al}_{2} \mathrm{O}_{3} / \mathrm{H}_{2} \mathrm{O}$ and water fluid between a two-phase turbulent flow. The results show that the effect of nanoparticles volume concentration in nanofluids reduces the growth process under the effect of heat transfer. The radius of vapour bubble in nanoparticles/water nanofluids is less than in pure water. The temperature distribution surrounding the regime of bubble growth in water is more intensive than in other cases of nanoparticles/water $\left(\mathrm{Al}_{2} \mathrm{O}_{3} / \mathrm{H}_{2} \mathrm{O}, \mathrm{Fe}_{3} \mathrm{O}_{4} / \mathrm{H}_{2} \mathrm{O},\right.$ and $\left.\mathrm{CuO} / \mathrm{H}_{2} \mathrm{O}\right)$ nanofluids. Besides, the given results demonstrate that the bubble radius in turbulent flow is less than in turbulent flow. The obtained results of bubble growth in the nanofluids are agreement with the results of previous studies [11, 21, 22, 24, 31] at special cases of turbulent flow and laminar flow. This conclusion must be taken into account in the field of the nuclear power system and other applications of bubbles field.