1 Introduction

Studying the heat transfer process is captivating to many engineers due to its wide range of applications like heat exchangers, fuel cell and electronic devices cooling constructions. As such, enhancement [development] of these processes can save energy and time by lowering the sizes of thermal systems.

To reach this goal, one approach is to enhance fluid flow characteristics—thermal conductivity and thermal diffusivity—which could be achieved by adding nano-size particles of metals, oxides, carbides and carbon to [a] base fluid. This late produced fluid is called nanofluid (Wong and De Leon 2010).

Solid particles have better thermal conductivity compared to conventional base fluids such as pure water, ethylene glycol or oil. The addition of solid nanoparticles increases the thermal conductivity of nanofluids (Abouali and Ahmadi 2012; Afshar et al. 2012). Copper’s (Cu) thermal conductivity is a good representative of a solid particle having greater thermal conductivity compared to water or engine oil as pure liquids (Chandrasekaran et al. 2014).

The concept of nanofluids was presented for the first time in 1995 by Choi (Eastman et al. 1995), while Pak and Cho (Pak and Cho 1998) were the first researchers to interpret the application of nanofluids in forced convection heat transfer and observe heat transfer coefficient enhancement of 45 and 75% with 1.34 and 2.78% Al2O3 particles in practice. They claimed that for [a] constant average velocity nanofluids, a 3–12% increase in heat transfer coefficient is obtained.

To attain information in calculating nanofluid heat transfer, a nano-layer can be mentioned (Tillman and Hill 2006). Studying non-equilibrium heat conduction in a nanofluid layer with periodic heat flux on one side and a specified temperature on the other side numerically shows that by adding nanoparticles to the base fluid, heat transfer increases (Zhang and Ma 2008).

Recent studies and experiments show that decreasing the size of particles would result in the increase of thermal conductivity. Four probable explanations can be presented, as no existing theory approves this: the particle’s Brownian motion; molecular liquid layers formation around the particles; heat transfer nature in the nanoparticle and nanoparticle clustering effect. Due to this observation the generally accepted diffusive transport mechanism is no longer valid at the nanoscale and that the ballistic transport is more realistic (Jang and Choi 2004; Keblinski et al. 2002).

The behavior of a building wall subjected to a Periodic heat, climatic temperature changes, was studied by Strub et al. (2005). The model used to investigate the problem was a solid homogeneous single-phase wall, subjected to a fluctuating external temperature and an ideally constant internal temperature to analyze the total entropy generation over time period and wall thickness.

Timofeeva et al. (2011) presented a summary of systematic experimental studies of both thermophysical and heat transfer in nanofluids. They claimed that the complexity and controversy of nanofluid systems are related to the boundary layers between nanoparticles and the base fluids, resulting in three-phase systems, instead of two-phase ones. They believed that this kind of consideration provides better understanding of the nanofluids.

The potentials of nanofluid application in solar energy systems, such as solar collectors, photovoltaic thermal systems and thermal energy storage systems have been indicated in multiple papers (Hassan et al. 2013; He et al. 2013; Lenert and Wang 2012; Sardarabadi et al. 2014; Yousefi et al. 2012; Zadeh et al. 2015).

Two main methods are utilized to study nanofluids. In the first method called single phase, the base fluid and particles are in thermal and hydrodynamic equilibrium, whereas in the second one, the two-phase model, the base fluid and particles behavior are studied individually. The first model is more popular with the analysts due to its relative simplicity.

The number of the studies using the two-phase modeling of nanofluids is low; therefore, the objective of this paper is to develop the two-phase modeling of studying nanofluids. The two-phase model is used for formulating the problem and an analytical solution is presented to solve it. Hereafter the non-equilibrium between nanoparticles and base fluid is studied by considering a number of physical quantities effect. In industries, when the operating conditions change, the temperature changes with time and fluctuating temperature problems are practical subject.

2 Problem Statement

In this paper, a nanofluid layer with initial temperature of T i and thickness L is considered. It is suddenly subjected to a fluctuating external temperature on one side and a specified temperature T B on the other side (Fig. 1).

Fig. 1
figure 1

Schematic representation of the layer and boundary conditions

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It is assumed that the layer is thin and free convection is negligible. When the boundary is heated, the heat is first transferred to the base fluid and then the nanoparticles are heated by it. In the physical aspect, there is a time delay between base fluid and particles temperatures.

The heat equation for nanoparticles can be written as:

$$ - \left( {\rho c_{\text{p}} } \right)_{\text{s}} V_{\text{s}} \frac{{\partial T_{\text{s}} }}{\partial t} = A_{\text{s}} h_{\text{s}} \left( {T_{\text{s}} - T_{\text{f}} } \right). $$
(1)

The subscripts s and f are used for fluid and nanoparticle quantity, respectively. The \( V_{\text{s}} ,A_{\text{s}} \) and \( h_{\text{s}} \) are total volume, total surface of nanoparticles and heat transfer coefficient at the nanoparticle surface, respectively.

In Eq. (1),

$$ A_{\text{s}} = nA^{\prime}_{\text{s}} ,v_{\text{s}} = nv^{\prime}_{\text{s}} . $$
(2)

The \( V^{\prime}_{\text{s}} ,A^{\prime}_{\text{s}} \) and n are nanoparticle volume, nanoparticle surface and total number of nanoparticles. The coupling factor G and volume fraction of nanoparticles φ are defined as:

$$ G = \frac{{\varphi A_{\text{s}} h_{\text{s}} }}{{V_{\text{s}} }}, $$
(3)
$$ \varphi = \frac{{V_{\text{s}} }}{{V_{\text{nf}} }} = \frac{{V_{\text{s}} }}{{V_{\text{s}} + V_{\text{f}} }}. $$
(4)

The subscript nf is used for nanofluid quantity. By using (2–4) in (1), the heat equation for nanoparticles is obtained as (Zhang 2006):

$$ \varphi \left( {\rho c_{\text{p}} } \right)_{\text{s}} \frac{{\partial T_{\text{s}} }}{\partial t} = G\left( {T_{\text{f}} - T_{\text{s}} } \right). $$
(5)

In Fig. 1 for the volume element of thickness dx, the following energy balance may be made which gives the heat equation for the base fluid:

$$ - \left( {\rho c_{\text{p}} } \right)_{\text{f}} dV_{\text{f}} \frac{{\partial T_{\text{f}} }}{\partial t} = nA_{\text{s}} h_{\text{s}} \left( {T_{\text{f}} - T_{\text{s}} } \right) + \frac{{\partial q_{\text{x}} }}{\partial x}dx, $$
(6)

in which:

$$ \begin{aligned} dV_{\text{f}} = & (1 - \varphi )dV_{\text{nf}} = (1 - \varphi )Adx, \\ q_{\text{x}} = & - k_{\text{nf}} A\frac{{\partial T_{\text{f}} }}{\partial x}, \\ n = & \frac{{\varphi dV_{\text{nf}} }}{{V_{\text{s}} }} = \frac{\varphi }{{V_{\text{s}} }}Adx, \\ \end{aligned} $$
(7)

where \( V_{\text{f}} \) is base fluid volume. Combining (6) and (7) gives

$$ - \left( {\rho c_{\text{p}} } \right)_{\text{f}} (1 - \varphi )\frac{{\partial T_{\text{f}} }}{\partial t} = \frac{\varphi }{{v_{\text{s}} }}A_{\text{s}} h_{\text{s}} \left( {T_{\text{f}} - T_{\text{s}} } \right) - k_{\text{nf}} \frac{{\partial^{2} T_{\text{f}} }}{{\partial x^{2} }}. $$
(8)

By using (3) in (8), the heat equation for base fluid is obtained as (Zhang and Ma 2008):

$$ \left( {1 - \varphi } \right)\left( {\rho c_{\text{p}} } \right)_{\text{f}} \frac{{\partial T_{\text{f}} }}{\partial t} = k_{\text{nf}} \frac{{\partial^{2} T_{\text{f}} }}{{\partial x^{2} }} + G\left( {T_{\text{s}} - T_{\text{f}} } \right). $$
(9)

The initial and boundary conditions for problem are:

$$ T_{\text{s}} \left( {x,0} \right) = T_{\text{f}} \left( {x,0} \right) = T_{i} , $$
(10)
$$ T_{\text{f}} \left( {0,t} \right) = T_{\text{i}} + (T_{\text{B}} - T_{\text{i}} )\lambda \sin \left( {\frac{2\pi t}{{t_{\text{p}} }}} \right),\quad atx = 0, $$
(11)
$$ T_{\text{f}} \left( {L,t} \right) = T_{\text{B}} ,\quad atx = L, $$
(12)

where \( \lambda \, \) and \( t_{\text{p}} \) are altitude and period of fluctuating temperature.

Some non-dimensional parameters are defined as (Zhang and Ma 2008):

$$ \begin{aligned} \theta = & \frac{{(T - T_{\text{i}} )}}{{(T_{\text{B}} - T_{\text{i}} )}};X = \frac{x}{L};\tau = \frac{{\alpha_{f} t}}{{L^{2} }}; \\ C_{\text{sf}} = & \frac{{(\rho c_{\text{p}} )_{\text{s}} }}{{(\rho c_{\text{p}} )_{\text{f}} }},K_{\text{eff}} = \frac{{k_{\text{nf}} }}{{k_{\text{f}} }},Sp = \frac{{GL^{2} }}{{\varphi k_{\text{f}} }}. \\ \end{aligned} $$
(13)

In these expressions, \( Sp \) is Sparrow number, \( k_{\text{eff}} \) and \( C_{\text{sf}} \) are called dimensionless effective thermal conductivity and heat capacity ratio, respectively.

By using non-dimensional parameter in heat equations, we have

$$ C_{\text{sf}} \frac{{\partial \theta_{\text{s}} }}{\partial \tau } = Sp(\theta_{\text{f}} - \theta_{\text{s}} ), $$
(14)
$$ (1 - \varphi )\frac{{\partial \theta_{\text{f}} }}{\partial \tau } = K_{\text{eff}} \frac{{\partial^{2} \theta_{\text{f}} }}{{\partial X^{2} }} + \varphi Sp(\theta_{\text{s}} - \theta_{\text{f}} ). $$
(15)

Using non-dimensional parameter in boundary and initial conditions yields:

$$ \theta_{\text{s}} \left( {X,0} \right) = \theta_{\text{f}} \left( {X,0} \right) = 0, $$
(16)
$$ \theta_{\text{f}} \left( {0,\tau } \right) = \lambda \sin \left( {\frac{2\pi \tau }{{\tau_{\text{p}} }}} \right), $$
(17)
$$ \theta_{\text{f}} \left( {L,\tau } \right) = 1. $$
(18)

For simplicity, in this paper we suppose the interfacial resistance of nanoparticle surface is negligible and in this special case \( k_{\text{eff}} \) can be formulated (Evans et al. 2006; MacDevette et al. 2013):

$$ k_{\text{eff}} = 1 + 3\varphi . $$
(19)

3 Analytical Solutions

The partial differential Eqs. (14) and  (15) are transformed to a system of ordinary differential equations by using sine transform definition.

The Fourier sine transform of the function \( \theta (X,\tau ) \) on interval [0, 1] is defined as (Kreyszig et al. 2011):

$$ F_{\text{s}} [\theta (X,\tau )] = 2\int\limits_{0}^{1} {\sin (n\pi X)\theta (X,\tau )dX = S_{\text{n}} (\tau )} . $$
(20)

Then \( \theta (X,\tau ) \) can be achieved as:

$$ \theta (X,\tau ) = \sum\limits_{n = 1}^{\infty } {S_{n} (t)\sin (n\pi X)} . $$
(21)

It is shown that:

$$ F_{\text{s}} \left( {\frac{{\partial^{2} (\theta ,\tau )}}{{\partial X^{2} }}} \right) = - \left( {\frac{n\pi }{L}} \right)^{2} S_{n} ( = \tau ) + \frac{2n\pi }{{L^{2} }}\left[ {\theta (0,\tau ) + ( - 1)^{n + 1} \theta (L,\tau )} \right]. $$
(22)

By using transform definition and its properties as shown above, the final heat equations and their conditions Eqs. (14)–(18) can be written as:

$$ C_{\text{sf}} \frac{{ds_{\text{n}}^{\text{s}} }}{d\tau } = Sp\left( {s_{\text{n}}^{\text{f}} - s_{\text{n}}^{\text{s}} } \right), $$
(23)
$$ \left( {1 - \varphi } \right)\frac{{ds_{\text{n}}^{\text{f}} }}{d\tau } = K_{\text{eff}} \left\{ { - (n\pi )^{2} S^{\text{f}}_{n} (\tau ) + 2n\pi \left[ {\lambda { \sin }\left( {\frac{2\pi \tau }{{\tau_{\text{p}} }}} \right) + ( - 1)^{n + 1} } \right]} \right\} + \varphi Sp\left( {s_{\text{n}}^{\text{s}} - s_{\text{n}}^{\text{f}} } \right), $$
(24)
$$ S_{\text{n}}^{\text{s}} (0) = 0,S_{\text{n}}^{\text{f}} (0) = 0. $$
(25)

The computation software Maple V.14 is used to solve the system of ordinary Eqs. (23) and  (24) with the boundary conditions (25) for some values of physical quantity, the functions of \( s_{\text{n}}^{\text{f}} \) and \( s_{\text{n}}^{\text{s}} \) are obtained. Then the final solution, dimensionless temperature distributions are obtained as:

$$ \theta_{\text{f}} (X,\tau ) = \sum\limits_{n = 1}^{j} {S_{\text{n}}^{\text{f}} (\tau )} \sin (n\pi X), $$
(26)
$$ \theta_{\text{s}} (X,\tau ) = \sum\limits_{n = 1}^{j} {S_{\text{n}}^{\text{s}} (\tau )} \sin (n\pi X). $$
(27)

The heat flux \( q^{\prime\prime} \) at any point \( x_{\text{a}} \) and the transferred heat \( q_{\text{m}} \) at any interval [\( m\tau_{\text{p}} ,(m + 1)\tau_{\text{p}} \)] can be achieved as:

$$ q^{\prime\prime}(t) = - k_{\text{nf}} \left. {\frac{{\partial T_{\text{f}} (x,t)}}{\partial x}} \right|_{{x_{\text{a}} }} , $$
(28)
$$ \frac{{q_{\text{m}} }}{A} = \int\limits_{{m\,t_{\text{p}} }}^{{(m + 1)t_{\text{p}} }} {q^{\prime\prime}(t)} {\text{d}}t. $$
(29)

By introducing dimensionless heat transferred parameter as:

$$ Q^{\prime}_{\text{m}} = \frac{{q_{\text{m}} LA}}{{k_{\text{nf}} \left( {T_{\text{B}} - T_{\text{i}} } \right)t_{\text{p}} }}, $$

and using non-dimensional parameters Eq. (13), Eq. (29) is rearranged as:

$$ Q^{\prime}_{\text{m}} = - \int\limits_{m}^{(m + 1)} {\left. {\frac{{\partial \theta_{\text{f}} (X,\tau /\tau_{\text{p}} )}}{\partial X}} \right|_{{X_{\text{a}} }} d(\tau /\tau_{\text{p}} )} . $$
(30)

The dimensionless total heat transferred at any point \( X_{\text{a}} \) for interval [\( 0,(m + 1)\tau /\tau_{\text{p}} \)] can achieve

$$ Q = \sum\limits_{m = 1}^{m + 1} {Q^{\prime}_{\text{m}} } . $$
(31)

4 Results and Discussion

For validity, the problem was solved numerically, 4th order Runge–Kutta, for the special case \( Sp = 500,\phi = 0.002,\tau_{\text{p}} = 0.01 \). Comparison showed that there is an excellent agreement between analytical and numerical results that is shown in Table 1. As mentioned in the previous section, the temperature distribution is achieved in several cases for special values of physical quantities. For all solved cases \( \lambda \) and \( C_{\text{sf}} \) values are selected 3 and 2.7, respectively. The solutions are shown graphically, because they are too long to be presented here. For reader’s physical consideration, spatial graphs are drawn for special cases \( (sp = 500,\tau_{\text{p}} = 0.01 \) and \( \,\varphi = 0.02) \) as shown in Fig. 2.

Table 1 Comparison of Analytical and numerical results
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Fig. 2
figure 2

Dimensionless temperature distributions for the special case \( (sp = 500,\tau_{\text{p}} = 0.01,\varphi = 0.02) \): a base fluid, b nanoparticle

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But to analyze the results, solutions should be presented in two-dimensional diagrams. In all the two-dimensional diagrams, continuous line and dashed symbol are used for base fluid and nanoparticle temperature distributions, respectively.

For \( (Sp = 500,\varphi = 0.02) \) the equations are solved at different values of \( \tau_{\text{p}} \) and Figs. 3 and 4 are prepared in order to investigate the effect of fluctuating period on response. It can be seen that by increasing \( \tau_{\text{p}} \), the non-equilibrium between the nanoparticles and base fluid decreases. In physical aspects when \( \tau_{\text{p}} \) increases (frequency decreases), the particles have more time to approach the base fluid temperature.

Fig. 3
figure 3

Dimensionless temperature distributions for different values of \( \tau_{\text{p}} \) at several X. a \( \tau_{\text{p}} \) = 0.01, b \( \tau_{\text{p}} \) = 0.02, c \( \tau_{\text{p}} \) = 0.05

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Fig. 4
figure 4

Dimensionless temperature distributions for different values of \( \tau_{\text{p}} \) at several \( \tau /\tau_{\text{p}} \). a \( \tau_{\text{p}} \) = 0.01, b \( \tau_{\text{p}} \) = 0.02, c \( \tau_{\text{p}} \) = 0.05

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For \( (\tau_{\text{p}} = 0.01,\varphi = 0.02) \) the equations are solved at different values of Sp and Fig. 5 is prepared in order to investigate the effect of Sparrow number on the response. It can be seen that the non-equilibrium between the nanoparticles and base fluid decreases with the increase of Sp. The sparrow number is proportional to \( h_{\text{s}} \) (the heat transfer coefficient at the nanoparticle surface), so by increasing Sp the heat transfer rate between nanoparticles and base fluid increases, contact resistances decreases and non-equilibrium effect is reduced. In the limiting case, when contact resistance approaches to zero, the non-equilibrium vanishes.

Fig. 5
figure 5

Dimensionless temperature distributions for different values of \( sp \) at several X. a Sp = 500, b Sp = 1000, c Sp = 2000

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For \( (\tau_{\text{p}} = 0.01,sp = 500) \) the equations are solved at different values of φ, volume fraction of nanoparticles and Fig. 6 are prepared in order to investigate the effect of volume fraction on response. It can be seen that the temperature fluctuating amplitude in the layer and non-equilibrium effect is increased when φ increases. When φ increases the number of nanoparticles in fluid is increased, as well as nanofluid conductivity and heat diffusion. It means that the heat penetration in nanofluids goes up and fluctuating boundary temperature has a bigger effect on the layer temperature distribution.

Fig. 6
figure 6

Dimensionless temperature distributions for different values of φ at several X. a φ = 0.01, b φ = 0.03, c φ = 0.05

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Besides, in all cases at a significant time after starting the process, the temperature of nanoparticles approached that of the base fluid at the constant temperature layer side. This behavior is not seen at the other side where the temperature fluctuates with time.

As an example, using Eq. (31) dimensionless total heat transferred Q is calculated and for special cases \( (sp = 500,\varphi = 0.02) \) at \( X_{\text{a}} = 0.5 \) is drawn in Fig. 7.

Fig. 7
figure 7

Dimensionless total heat transfer behavior

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It can be seen that Q approaches to some value and remains constant after a couple of fluctuating. It means that after an adequate time, temperature fluctuates in a way that the heat flowing on the right side is equal to the one on the left side inside the two halves of a cycle. It can be seen that by increasing \( \tau_{\text{p}} \) the approached value decreases.

5 Conclusions

In this article, an analytical method is offered for solving the partial differential equations of a nanofluid layer which is formulated by the two-phase model. The layer is subjected to a fluctuating external temperature on one side and a specified temperature on the other side. The non-dimensional temperature distribution for base fluid and nanoparticles is obtained. The results show that the non-equilibrium between the nanoparticles and base fluid decreases with the increase of the fluctuating temperature’s period, the volume fraction of nanoparticles and the heat transfer coefficient at the nanoparticle surface.