1 Introduction

Melt flow on walls is an intrinsic part of the solidification process, e.g. melt flow along the mold surface during casting. Flow of melt has two positive effects, one is the heterogeneous nucleation on the cooling surface, and the other is that the heterogeneous nucleation sites can be swept into mold due to the flow of melt. These two positive effects are common, especially in the process of casting in metal molds [1, 2]. The dynamic heat transfer and cooling of melt flow near surfaces are important for the formation of casting microstructures. Eskin [3] researched the effect of ultrasonic oscillation on the cooling of melt, and it was found that a number of cavitation bubbles could be induced by ultrasonic oscillation. The cavitation bubble could collapse instantaneously, and shock waves were formed which could lead to partially high temperature and high pressure at the same time, so that the cooling rate decreased. Zocchi [4] found that velocity gradient existed in melt, and it led to inner friction when melt was ultrasonicated. So, the energy of ultrasonic oscillation could be transferred to internal energy of melt, and the cooling rate of melt decreased. The effect of mechanical vibration on the cooling rate of aluminum alloy melt was researched by Omura [5], and the results showed that the cooling rate increased with increasing vibration frequency. The mechanism of this effect has two explanations. One is that the gas between melt and mold surface is pressurised and the gas gap decreases due to mechanical vibration, so the contact area increases as well as the cooling rate [6]. The other explanation is that, the solidified layer near the mold surface can be broken by mechanical vibration, and there are increase in contact area and cooling rate [7]. However, relatively little research about dynamic heat transfer and cooling of melt flow on vibrating wall exists, and the mechanism is not clear. The dynamic heat transfer and cooling of melt flow on vibrating wall is a research topic in casting that needs to be explored.

A vibrating plate is a convenient, low cost method of melt treatment [8,9,10,11]. Overheated melt is cast onto the vibration cooled inclined plate, and a number of heterogeneous nucleation sites are induced on the surface of the plate. The heterogeneous nucleation sites are swept into the mold at the exit of the plate under the function of flow and vibration, and finally the solidification microstructure composed of fine spherical grains are formed [8]. The flow of melt on a vibrating cooling slope is a solidification process under the coupled effects of flow and vibration, and it is a new method to investigate the effect of dynamic heat transfer and cooling of melt flow on vibration wall. Based on previous research, a model of dynamic heat transfer and cooling of melt flow on vibration wall has been established, and verification experiment has been carried out in this paper, which can provide theoretical guidelines to further research.

2 Heat Transfer Model of Melt Flow on Vibration Slope

Figure 1 is the geometric model of melt flow on a vibrating inclined plate, and the vibration direction of the slope is parallel to the flow direction of the melt, in which H is casting height, θ is plate angle, f is vibration frequency, and A is vibration amplitude. In order to describe the physical process of melt flow on a vibrating plate, two assumptions have been made based on previous research. One is that, the plate is exposed to a sinusoidal vibration, and another is that the vibrations of the slope and the melt are synchronous.

Fig. 1
figure 1

Geometric model of melt flow on a vibrating inclined plate

Full size image

The formula of Reynolds number Re of melt flow on inclined plate without vibration [12] is \(\text{Re} = \frac{{u_{\infty } x}}{v}\), in which u∞ is flow velocity and it is described as \(u_{\infty } = (2gH\sin^{2} \theta + 2gx\sin \theta )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}\), g is the acceleration due to gravity, v is the viscosity of melt, and x is the flow length. According to Ref. [13], the vibration Reynolds number of melt induced by sinusoidal vibration of the inclined plate can be defined as \(\text{Re}_{\text{v}} = \frac{2\pi fAx}{v}\), so Reynolds number of melt flow on vibrating slope is \(\text{Re}_{x} = \text{Re} + \text{Re}_{\text{v}} = \frac{x}{v}(u_{\infty } + 2\pi fA)\).

Figure 2 is the heat transfer diagram of melt on vibrating inclined plate, in which T is the temperature of the melt, δ is the temperature boundary layer thickness of the melt, Tbs is the temperature of the upper surface of the plate, Tbx is the temperature of the underside of the plate, and Tw is the temperature of the cooling water. There are three modes of heat loss of melt on the plate: convective heat transfer between the melt and slope, convective heat transfer between the melt and air, and heat radiation of the melt. The heat loss caused by convective heat transfer between the melt and the plate is much higher than the heat loss induced by the other two processes. So, convective heat transfer between the melt and air and heat radiation of melt can be ignored for the convenience of research on the heat transfer of melt [14]. In this paper, only the convective heat transfer between melt and plate has been taken into consideration. The heat transfer between the melt and plate can be divided into three stages: heat transfer from melt to the upper surface of the slope by convection, heat transfer from the upper surface to the lower surface by conduction, and heat transfer from under surface to cooling water by convection. During the stable process of heat transfer, the amounts of transferred heat are equal in the three stages.

Fig. 2
figure 2

Heat transfer diagram of melt on a vibrating inclined plate

Full size image

2.1 Distribution of Temperature Boundary Layer Thickness in Melt on Vibration Slope

The temperature boundary layer thicknesses [12] of laminar flow and turbulent flow on a plate without vibration can be described as \(\delta_{vl} = \frac{5x}{{\text{Re}_{x}^{1/2} \hbox{Pr}^{1/3}} }\) and \(\delta_{vt} = \frac{0.376x}{{\text{Re}_{x}^{1/5} }}\), respectively. Replace the Reynolds number of melt in the equations to get the temperature boundary layer thicknesses of laminar flow and turbulent flow on a vibrating plate, as is shown in Eqs. (1) and (2), in which Pr is prandtl number, and the prandtl number of metal melt is between 0.0004 and 0.029 [14].

$$\delta_{l} = 5\hbox{Pr}^{-1/3} \sqrt {\frac{vx}{{(2gH\sin^{2} \theta + 2gx\sin \theta )^{1/2} + 2\pi fA}}}$$
(1)
$$\delta_{t} = 0.376\left[ {\frac{{x^{4} v}}{{(2gH\sin^{2} \theta + 2gx\sin \theta )^{1/2} + 2\pi fA}}} \right]^{1/5}$$
(2)

A356 alloy is used as the experiment material, and its composition is shown in Table 1. Setting ν = 4.82 × 10−7 m2/s, θ = 45° and H = 0.1 m, we get the effect of different vibration parameters as shown in Fig. 3. Figure 3a shows the effect of different vibration frequencies on the distribution of temperature boundary layer thickness when vibration amplitude is 1 mm. It can be seen that there is a transition in the temperature boundary layer thickness when melt transfers from laminar flow to turbulent flow. The temperature boundary layer thickness of the laminar and turbulent flow increases with the increase of flow length, and it decreases with the increase of vibration frequency. Figure 3b shows the effect of different vibration amplitudes on the distribution of temperature boundary layer thickness when vibration frequency is 70 Hz. It can be seen that temperature boundary layer thickness decreases with the increase of vibration amplitude. The reason is that, the disturbance in melt induced by vibration can accelerate the convective heat transfer between the melt and slope and increase the velocity gradient in melt.

Table 1 Composition of A356 alloy
Full size table
Fig. 3
figure 3

Distributions of temperature boundary layer thickness of melt along flow direction at different vibration frequencies and amplitudes. a Amplitude is 1 mm, and vibration frequencies are different; b Vibration frequency is 70 Hz, and vibration amplitudes are different

Full size image

2.2 Convection Coefficient Between Melt and Vibration Slope

Equation (3) can be obtained according to the definition of Nusselt number and its empirical formula [12, 15], in which Nu is Nusselt number of metal melt, \(\bar{h}\) is the average convection coefficient between melt and slope, d is the characteristic length of heat transfer surface, and λ is the coefficient of heat conductivity of melt. C and n related to Re are empirical parameters, and the relationship between them is shown in Table 2. In this paper, d is the length of the plate. Based on Eq. (3), the average convection coefficient between the melt and slope without vibration can be described as \(\overline{h} = 1.1C\text{Re}_{x}^{n} \hbox{Pr}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} \frac{\lambda }{L}\), so the average convection coefficient between the melt and slope with vibration is shown in Eq. (4).

$$Nu = \frac{{\bar{h}d}}{\lambda } = 1.1C{\text{Re}}_{x}^{n} \hbox{Pr}^{1/3}$$
(3)
$$\bar{h} = 1.1C\left({\frac{{\overline{{u_{\infty } }} x + 2\pi fAx}}{v}} \right)^{n} \hbox{Pr}^{1/3} \frac{\lambda }{L}$$
(4)
Table 2 Values of C, n and Re [12]
Full size table

Setting λ = 233 J/(m s K), L = 0.5 m, θ = 45° and H = 0.1 m, we get the distributions of average convection coefficient between A356 melt and slope at different vibration parameters and are shown in Fig. 4 based on Eq. (4). Figure 4a shows the effect of different vibration frequencies on the distribution of average convection coefficient when vibration amplitude is 1 mm. It can be seen that the average convection coefficient between A356 melt and plate increases with the increase of flow length along slope, and increases with the increase of vibration frequency. Figure 4b shows the effect of different vibration amplitudes on the distribution of average convection coefficient when vibration frequency is 70 Hz. It can be seen that average convection coefficient increases with the increase of vibration amplitude.

Fig. 4
figure 4

Distributions of average convection coefficient between melt and slope along flow direction at different vibration frequencies and amplitudes. a Amplitude is 1 mm, and vibration frequencies are different; b Vibration frequency is 70 Hz, and vibration amplitudes are different

Full size image

2.3 Convection Coefficient Between Cooling Water and Vibration Slope

When the cooling slope does not vibrate, the average convection coefficient between laminar flow cooling water and plate can be expressed as \(\overline{{h_{w} }} = \frac{{0.664\lambda_{w} \hbox{Pr}_{w}^{1/3} \hbox{Re}^{1/2} }}{x}\), while the average convection coefficient between turbulent flow cooling water and plate is \(\overline{{h_{w} }} = \frac{{0.0592\lambda_{w} \hbox{Pr}_{w}^{1/3} \text{Re}^{4/5} }}{x}\), in which \(\overline{{h_{w} }}\) is the average convection coefficient between cooling water and slope, λ w is the heat conductivity coefficient of cooling water, Pr w is Prandtl number of cooling water, and vw is the viscosity of cooling water. λ w is about 0.5 W/(m K), and Pr w is about 7 at room temperature [12]. So the average convection coefficient between laminar flow cooling water and vibration slope is shown in Eqs. (5) and (6) shows the average convection coefficient between turbulent flow cooling water and vibration plate.

$$\overline{{h_{w} }} = 0.664\lambda_{w} {\hbox{Pr}}_{\text{w}}^{1/3} \left({\frac{{u_{w} + 2\pi fA}}{{xv_{w} }}}\right)^{1/2}$$
(5)
$$\overline{{h_{w} }} = \frac{{0.0592\lambda_{w} {\hbox{Pr}}_{w}^{1/3} }}{{x^{1/5} }}\left( {\frac{{u_{w} + 2\pi fA}}{{xv_{w} }}} \right)^{4/5}$$
(6)

The relationship of average convection coefficient between cooling water and slope with different vibration parameters and flow velocity of cooling water is shown in Fig. 5 according to Eqs. (5) and (6) when θ is 45° and H is 0.1 m. Figure 5a shows the effect of flow velocity of cooling water on average convection coefficient when vibration amplitude is 1 mm and vibration frequencies are different. It can be seen that the average convection coefficient between cooling water and slope increases with increasing flow velocity of cooling water and increasing vibration frequency. Figure 5b shows the effect of flow velocity of cooling water on average convection coefficient when vibration frequency is 70 Hz and vibration amplitudes are different. It can be seen that average convection coefficient increases with the increase of vibration amplitude. The reason is that, vibration also can induce disturbance in cooling water, increase Reynolds number of cooling water, and accelerate the transition from laminar to turbulent flow of the cooling water.

Fig. 5
figure 5

Relationship between average convection coefficient between cooling water and slope and the velocity of cooling water at different vibration frequencies and amplitudes. a Amplitude is 1 mm, and vibration frequencies are different; b Vibration frequency is 70 Hz, and vibration amplitudes are different

Full size image

3 Cooling Rate of Melt on Vibration Slope

Based on the temperature compensation method and ignoring the convective heat transfer between melt and air and heat radiation from the melt, the cooling rate of the melt on the slope without vibration is \(\frac{\Delta T}{\Delta t} = \frac{{\bar{h}(T_{l\infty } - T_{bs} )}}{{\rho h_{y} C_{P} }}\), in which h y is the thickness of melt, and C p is the specific heat of melt [14]. So, the cooling rate of melt on vibration plate is shown in Eq. (7).

$$\frac{\Delta T}{\Delta t} = \frac{{1.1C\text{Re}_{x}^{n} \hbox{Pr}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} \lambda \left( {T_{l\infty } - T_{bs} } \right)}}{{L\rho h_{y} C_{p} }}$$
(7)

If we set the specific heat of A356 alloy melt to 1050 J/(kg K), the distribution of the temperature of the melt on the vibration slope with different vibration parameters along flow direction are shown in Fig. 6 based on Eq. (7) when L is 0.5 m, h y is 2 mm, H is 0.1 m, θ is 45°, casting temperature is 700 °C, and T bs is 30 °C. Figure 6a shows the cooling rate distribution of the melt along the flow direction when the vibration amplitude is 1 mm and vibration frequency varies. It can be seen that the cooling rate increases with the increase of flow length and the increase of vibration frequency. Figure 6b shows the cooling rate of the melt along the flow direction when the vibration frequency is 70 Hz and the vibration amplitudes vary. It can be seen that cooling rate increases with the increase of vibration amplitude. The cooling rate can reach 400-600 K/s which belong to the class of sub-rapid solidification. During the whole flow process, there is no disturbance of cooling rate when the flow style transfers from laminar flow to turbulent flow. So vibration not only increases the cooling rate, but also stabilizes the heat transfer between melt and cooling slope.

Fig. 6
figure 6

Distributions of cooling rate of alloy melt along flow direction at different vibration frequencies and amplitudes

Full size image

4 Verification Experiment

Figure 7 shows the solidification microstructure after flow on the plate with different vibration parameters when H is 0.1 m, θ is 45° and h y is 2 mm. The microstructure is composed of dendrites and spherical grains. When the plate does not vibrate, dendrite grain is in the majority and the average grain size is about 100 μm, as Fig. 7a shows. When the vibration frequency is 30 Hz and the vibration amplitude is 1 mm, the size of the dendrite grains decreases, and the proportion of dendrite grain decreases markedly with the average grain size of about 70 μm, as shown in Fig. 7b. The grain size is much smaller and the proportion of dendrite grain is much less when the vibration frequency is 100 Hz and vibration amplitude is 3 mm, as Fig. 7c shows. It is thus found that the grain size decreases with the increase of vibration intensity. What’s more, with the increase of vibration intensity, the nucleation rate of melt flow on the slope increases and the cooling rate of melt increases too. So, the effects of vibration intensity on cooling rate of melt on the slope resulting from the established model and verification experiment agree with each other, and the established model in this paper can relatively well explain the heat transfer and cooling rate of alloy melt flow on vibration plate.

Fig. 7
figure 7

Solidification microstructures of melt after flow on slope with different vibration parameters. a No vibration; b Vibration frequency is 10 Hz and vibration amplitude is 1 mm; c Vibration frequency is 100 Hz and vibration amplitude is 3 mm

Full size image

5 Conclusion

In this paper, a model for heat transfer of melt flow on a vibration surface is established, and the coupling effects of vibration and shear on heat transfer and cooling rate of melt on plate are analyzed. The calculations are based on A356 alloy melt and the conclusions are as follows: With the increase of flow length, the temperature boundary layer thickness increases in both laminar flow melt and turbulent flow melt on the slope. With the increase of vibration frequency and amplitude, the temperature boundary layer thickness in the melt on the plate decreases. The reason is that, the disturbance of the alloy melt induced by vibration can strengthen the heat transfer between melt and slope, and increase the temperature gradient in melt. With the increase of length along flow direction, vibration frequency and vibration amplitude, the heat transfer coefficient between the alloy melt and plate, heat transfer coefficient between the cooling water and slope, and cooling rate of melt increases. The cooling rate of alloy melt on the plate can reach 400–600 K/s when the flow length is 0.5 m, which belongs to the class of sub-rapid solidification. The heat transfer type does not change during the whole flow process of alloy melt on the slope, so vibration can not only strengthen the cooling rate of melt, but also stabilize the heat transfer between melt and plate. The solidification microstructures variant as a function of different vibration parameters are gained by experiment. The grain size decreases with increasing vibration intensity, which shows that vibration can increase cooling rate and nucleation rate. So, the effects of vibration on heat transfer and cooling rate obtained from the established model and verification experiment agree with each other, and the model established in this paper can relatively well explain the heat transfer and cooling rate of alloy melt flow on a vibrating inclined plate.