1 Introduction

Spray rolling is a novel metallic semisolid near-net-shape forming technique. It was proposed by E.J. Lavernia and K.M. McHugh in 2003 and may be traced back to A.R.E. Singer’s pioneering work in the 1970s.[1,2] Spray rolling combines the advantages of twin-roll casing and spray forming in a single operation and shows promise for providing a new method of plate and strip production resulting in energy and materials saving, a short production run, and high-performance process for the metal industry.[3–5] In spray rolling (Figure 1), the molten aluminum alloy metal stream is atomized into disintegrated, microsized droplets by an inert gas and deposited on the roller surface. Then, the deposited material is dragged into the roll gap and consolidated to a fully dense strip. The spray-rolling process has been under development in few industrial countries.[6–8]

Fig. 1
figure 1

Schematic of spray rolling [1]

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By controlling the process properly, especially the deposit shape and thickness, it is possible to control spray rolling yield in the manufacture of near-net-shape preforms of strips such as 7050 aluminum alloy. An inspection of the available literature demonstrates that the shape models have been developed and applied to describe the deposit shape evolution during spray forming process by dealing with a continuous droplet mass flux or individual droplets.[9–12] Moreover, it is evident that the droplet splashing, redeposition of materials, and topological features of deposition surface are important factors in governing the deposit shape.[13,14] However, it should be noted that the previous work on spray forming focused on deposits only in the shape of a billet, tube, or ring.[15] In spray rolling, two deposition surfaces are arranged closely next to each other. The rebounding of droplets from the surface of one wheel to another, namely the droplet rebounding and redeposition phenomenon, an important factor in governing the deposit shape, should be considered. The dynamics of the spray-rolling process presents a significant challenge for the development of meaningful models. Lin et al.[4,5] studied the transient to steady-state transition and the selection of the spray deposition rate during the spray-rolling process. However, little information has been obtained for the deposit shape evolution.

This article provides a theoretical analysis of deposit shape evolution during spray rolling by considering the geometrical features of deposition surface and the droplet rebounding and redeposition phenomenon. We focus on determining the entry angle for spray-rolling process in order to obtain a continuous operation. The simulation results can be compared with the experimental results via deposit thickness and strip production rate.

2 Modeling

2.1 Mass Flux Distribution in the Spray Cone

In the linear atomizer nozzle, the slit for the molten metal delivery is parallel to the roll axes. A cross-sectional geometry normal to the roll axes is used to represent the overall geometry of the spray-rolled material, as shown in Figure 2(a). In order to deal with the complicated geometrical features of deposition surface, a function of the spray following a Gaussian distribution was used.[15]

Fig. 2
figure 2

(a) Relative geometrical position of the spray cone to the roller wheel. (b) A schematic illustration of droplet scattering and redeposition

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$$ m(r,h) = m_{\max 0} \cdot \left( {\frac{{h_{0} }}{h}} \right)^{2} \exp\left ( - a_{0} \cdot \left( {\frac{{h_{0} }}{h}} \right)^{k} r^{k}\right ) $$
(1)

where k is a constant, r presents the width of the spray cone, and a 0 is a radial distribution coefficient. Once the maximum mass flux in the spray cone m max0 (kgm−2s−1) at a reference spray distance h 0 is known, the distribution of mass flux m(r, h) at any other position can be calculated. For deposition onto the surface of a rotating mandrel with a radius R at a rotational speed ω, the coordinate transformation of an arbitrary point (r, h) can be expressed as: r = x + δ/2 + R; h = H + R – y, where H is the spray distance and δ is the distance between two rollers.

2.2 Sticking Efficiency

When gas-atomized droplets impinge the substrate or the deposit, a part of droplets was splashed or rebounded away from the deposition surface. Usually, sticking efficiency SE is defined to express the sticking ability of the impinging droplets. This coefficient depends on several factors: the residual liquid fraction and kinetic energy of the impinging droplets, the impact angle, the thermal conditions of the deposition surface, and the interaction between the deposition surface and the droplets. For deposition onto the surface of a rotating mandrel where the geometrical effect is a dominant factor for droplet deposition, the sticking efficiency in spray rolling should relate to the droplet impact angle at deposition. Mathur et al.[16] proposed the sticking efficiency SE which included both thermal sticking efficiency SE(T) and geometrical sticking efficiency SE(θ).

$$ SE = SE(T) + SE(\theta ) $$
(2)

where T is the temperature and θ is the impact angle between the incident mass and the surface normal at the point of deposition, e.g., θ 1 in Figure 2(b). For simplicity, a mean sticking efficiency SE(T) = 0.76 was selected to avoid the coupling of the thermal analysis with the shape model.[17,18] Based on consideration of the geometrical effect of a rotating mandrel on droplet deposition, a geometrical sticking efficiency SE(θ) = 0.76 × cos θ was assumed for 7050 aluminum alloy.[16]

2.3 Deposition, Redeposition, and the Deposit Growth in Thickness

As shown in Figure 2(b), when gas-atomized droplets impinge the substrate, a fraction of the arriving mass is retained on the surface at the point of impact, and it contributes to the growth of the surface. The primary deposition mass flux m 1 at a point (r, h) is given by

$$ m_{1} = g \cdot SE \cdot m(r,h) $$
(3)

where g is a coefficient defined as a visibility index. The considered substrate surface faces to the spray directly; thus, g is set to 1. Rebounding is found to be important for large solid droplets. The scattered mass flux from the deposition point is m s. Mass conservation gives

$$ m_{s} = \left( {1 - SE} \right) \cdot m_{i} (r,h) $$
(4)

The scattered mass either redeposits on the surface or is lost as overspray. As shown in Figure 2(b), the scattering mass flux m 2s was assumed to be distributed with the scattering angle θ 2 and governed by a mass scattering function B (θ i ). The total redeposition mass flux m 2 over the entire deposition surface ds is given by

$$ m_{2} = \frac{1}{ds} \cdot \int\limits_{\Upomega } {\left( {1 - SE} \right) \cdot m(r,h) \cdot B\left( {\theta_{i} } \right)} \cdot d\Upomega $$
(5)

where Ω denotes the deposition surface.

The deposition of individual droplets, initially on the surface of rollers and subsequently on the previously deposited material’s surface, is a sequence of discrete events. The formation of deposit shape occurs by successive accumulation of discrete droplets. The deposit growth in thickness ΔT thick is given by

$$ \Updelta T_{\text{thick}} = \left( {g \cdot SE \cdot m_{1} + m_{2} } \right) \cdot \frac{1}{\rho } \cdot \Updelta t $$
(6)

where ρ is the density of deposited materials.

2.4 Numerical Algorithm

We can calculate the deposit shape by the method of “curve tracking.” As shown in Figure 2(a), a tangent line from the atomizer to the roller determines the limiting positions for deposition onto the roller surface. The tangent point A(x 0 , y 0) corresponds to the starting angle θ 0 of the spray deposition

$$ \theta_{0} = {\pi} - { \arctan }\left( {\frac{H + R}{{\left( {\delta /2 + R} \right)^{0.5} }}} \right) - { \arccos }\left( {\frac{R}{{\left( {\left( {R + \delta /2} \right)^{2} + \left( {H + R} \right)^{2} } \right)^{0.5} }}} \right) $$
(7)

Once spray rolling is initiated, the position changes from point A to point C in time Δt, as shown in Figure 3(a). This can be divided into two processes that occur simultaneously. The position moves from A(x 0 , y 0) to B(x x , y y ) due to thickening of the deposited materials with arrival of the atomized droplets at roller surface, where y y = y 0 + ΔT thick and x x  = x 0 , the angle Δθ grow in every time moment Δt is given by: Δθ grow = arctan(y y / x x ) – arctan (y 0/x 0). Then, the position B(x x , y y ) moves to C(x end , y end) due to rotation of the roller surface, the angle Δθ rot in every time moment Δt is given by: Δθ rot = ωΔt. Therefore, the coordinates of point C (x end, y end) is given by

Fig. 3
figure 3

(a) A method of “curve tracking.” (b) Schematic diagram for the rejecting force and drag-in force

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$$ \begin{gathered} y_{\text{end}} = \, (y_{0} + \Updelta T_{\text{thick}} ) \, \sin \, (\theta_{0} + \Updelta \theta_{\text{grow}} + \Updelta \theta_{\text{rot}} )/ {r} \sin (\theta_{0} + \Updelta \theta_{\text{grow}} ) \hfill \\ x_{\text{end}} = x_{0} \cos \, (\theta_{0} + \Updelta \theta_{\text{grow}} + \Updelta \theta_{\text{rot}} )/\cos \, (\theta_{0} + \Updelta \theta_{\text{grow}} ) \hfill \\ \end{gathered} $$
(8)

At each computational time step, the rotational angle variation Δθ in time Δt is equal to the sum of Δθ grow and Δθ rot. The mass fluxes associated with droplet primary deposition and redeposition are given by Eqs. [3] and [4], the growth of each surface patch over a time step is determined through a numerically integrating Eq. [6]. Then the coordinates of the deposition surface during spray rolling are obtained, updated and tracked over all the computational time steps.

2.5 Determination of Drag-In Angle

As a result of thickening, the material deposited on the two roller surfaces overlaps eventually at point D(x drag, y drag), as shown in Figure 3(b). The drag-in angle θ drag is determined as

$$ \theta_{\text{drag}} = arctan\left( {y_{\text{drag}} /\left( {R^{2} - y_{\text{drag}}^{2} } \right)^{0.5} } \right) $$
(9)

As shown in Figure 3(b), the overlapping of the deposited material at point D means the beginning of a steady state (the geometry of the deposited materials on the roller surface maintaining its top surface profile) during the spray-rolling process. When the vertical component of the radial rolling force N y (θ drag Ncosθ, where N is the radial rolling force) is larger than the vertical component of the friction force P y (Psinθ), the metal can be dragged into the rollers successfully. In other words, the drag-in angle should not exceed the maximum value θ max, where θ max = tan−1 μ. 7050 aluminum alloy is chosen as the material model in this work. The friction coefficient μ between the roller and Al alloy is 0.2.[19] Thus, the maximum drag-in angle (θ max) is 0.1972 radians.

2.6 Determination of Production Rate

Under steady-state conditions, as shown in Figure 2(b), the mass conservation during the spray-rolling process can be described by[20]

$$ M = \rho_{\text{drag}} h_{\text{drag}} v_{\text{drag}} = \rho_{\text{neu}} h_{\text{neu}} v_{\text{neu}} $$
(10)

where M is the production rate (kgs−1 per meter of nozzle length) and v (ms−1) is the velocity of the rolled materials along the rolling direction. h neu and h drag are the thickness of the spray-rolled materials at the rolling neutral plane and entry plane, respectively. The value of h neu is given by h neu = D – D × cosθ neu + δ, where θ neu is the neutral angle. The value of v neu is equal to that of the rollers, v neu = R × ω × cos θ neu. In the case of the spray-rolling process, Bland’s methodology is still valid[4,21]; i.e., the radial rolling pressure at the side from the rolling entry plane to the neutral plane is equal to that at the side from the neutral plane to the rolling exit plane, which leads to the following equation for the neutral angle:

$$ \theta_{\text{neu}} = \sqrt {\frac{\delta }{R}} { \tan }\left[ {\frac{1}{2}{ \tan }^{ - 1} (\sqrt {\frac{R}{\delta }} \theta_{\text{drag}} ) - \frac{1}{2\mu }\ln (\frac{{h_{\text{drag}} }}{\delta })\sqrt {\frac{\delta }{R}} } \right] $$
(11)

If the drag-in angle and the thickness of the spray-rolled material in the entry plane are known, then the strip production rate can be calculated.

3 Result and Discussion

3.1 Droplet Scattering and Redeposition

For deposition of 7050 aluminum alloy on two tool steel mandrels with a diameter of 0.2 m, a fixed roll gap of 0.004 m, rolling at 3.14 rps and a spray distance of 0.4 m, the calculated ΔT thick in a time moment Δt conforms to different Gaussian distributions, as shown in Figure 4(a). In this case, the value of m max0 is determined as 50 kgm−2s−1 at a referenced spray distance of 0.4 m. The distribution of ΔT thick on a rotating mandrel expands a lower peak at the center of the deposition region compared with that on a flat substrate. By comparing the primary deposition only (m 1) with the primary deposition plus redeposition (m 1 + m 2), the droplet redeposition contributes 16 pct to the ΔT thick by calculation. It is evident that the geometrical features of the deposition surface and the droplet redeposition of materials are important factors in governing the deposit shape. The contribution of Δθ grow on Δθ at different deposition positions is demonstrated in Figure 4(b). When the value of x is greater than 0, Δθ grow means positive, if not, Δθ grow means negative. However, the effect of Δθ grow on Δθ is small compared with that of Δθ rot.

Fig. 4
figure 4

Spatial distribution of ΔT thick at a spray distance of 0.4 m (a) and the angle Δθ grow (b) in a time moment Δt on a roller surface. (c) Comparison of simulated and experimental results of the deposit cross-sectional final shape

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The cross-sectional final shapes of a deposit under a steady state obtained by experiments and simulation are shown in Figure 4(c). It can be seen that the calculated cross-sectional final shape of a deposit using the deposition plus redeposition model agrees well with that by experiments. Compared with the calculated value of h m1 0.0049 m taking no account of any redeposition, the calculated value of h m1+m2 taking account of redeposition is 0.0059 m, which matches closely the measured value of h measured 0.0065 m by experiments. Also, the calculated value of h m1+m2 consists well within the range of h min 0.0054 to h max 0.0079 m (i.e., the range of y min 0.01184 to y max 0.01959) obtained by McHugh et al. under the same conditions in Reference 4. The deviation between the simulated and experimental contours at the center of the deposition region is due to the experimental limitations. In the spray-rolling process, spray cannot be stopped simultaneously. The substrate receives additional deposits, and thus, the deposit thickness increases in the experiment.

Moreover, the experimental results can be used to verify the analytic model by strip production rate. Compared with the calculated value of M 3.64 kgs−1per meter of nozzle length taking no account of any redeposition, the calculated value of M considering redeposition is 3.69 kgs−1 per meter of nozzle length, which agrees with the measured value of M 3.72 kgs−1 per meter of nozzle length by experiments. The calculated values agree with the experimental data very well. At present, the production rates reported during the spray-rolling process can reach approximately 2.5 to15 kgs−1 per meter of nozzle length.[4,22]

3.2 Influence of Processing Parameters on Deposit Shape Evolution

The method described above was applied in a series of examples to demonstrate the effect of the processing parameters on deposit shape evolution under a steady state. According to the experimental conditions, the typical ranges of processing parameters considered in the simulation are as follows: maximum mass flux in the spray cone m max0 ranges from 30 to 90 kgm−2s−1, gap between the two mill rollers δ ranges from 0.002 to 0.005 m, roller diameters D ranges from 0.2 to 0.35 m, rotational frequency of rollers ω ranges from 2.5 to 4 rps, and spray distance H ranges from 0.35 to 0.5 m. The rotational angle varies from θ 0 to π. From Figure 5 to Figure 9, it is found that the deposit thickness and size increase gradually with rotational angle.

Fig. 5
figure 5

Effect of maximum mass flux in the spray cone on the deposit thickness

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The maximum mass flux in the spray cone has a significant influence on the deposit thickness, as shown in Figure 5. The deposited material with a maximum mass flux of 30kgm−2s−1 cannot be observed to overlap and contact each other. When the maximum mass fluxes in the spray cone are 50 and 90 kgm−2s−1, the values of the drag-in angle at which the materials deposit on the surface of two rollers overlap to contact each other are 0.1354 radians and 0.2563 radians, respectively. Obviously, the deposit thickness with a large mass flux is relatively thick compared with that with a small melt maximum mass flux. However, in the case of 90 kgm−2s−1 maximum mass flux, the deposited materials can be observed to overlap and contact each other but cannot be pulled into the gap of two rollers because the drag-in angle does not exceed the maximum drag-in angle θ max of 0.1972 radians. The maximum mass flux in the spray cone should be selected in the range of 50 kgm−2s−1 to 70 kgm−2s−1. The production rate for a maximum mass flux of 70 kgm−2s−1 increases to 3.08 kgs−1per meter of nozzle length compared with 2.98 kgs−1 per meter of nozzle length for a maximum mass flux of 50 kgm−2s−1.

Figure 6 shows the effect of spray distance on the deposit thickness when other processing parameters are kept constant. By changing the spray distance from 0.35 to 0.5 m, the variation of deposit shape and dimension is obvious. Results suggest that the deposit thickness decreases with increasing spray distance. This is attributed to the fact that the mass flux distribution for a short spray distance is very concentrated, and most droplets hit the surface of rollers directly and only a small portion sprays off the surfaces of rollers. With increasing spray distance, excessive droplets cannot reach the substrate surface, thus the thickness is reduced. The deposited material with a spray distance of 0.35 m can be dragged into the gap between two rollers. The production rate is 3.07 kgs−1 per meter of nozzle length.

Fig. 6
figure 6

Effect of spray distance on the deposit thickness

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The deposit thickness profiles for different roller diameters are illustrated in Figure 7. The larger the roller diameter, the thinner the deposit thickness. The small diameter of a mandrel can cause a large amount of droplet overspray, which can make the deposited material thinner. However, the linear speed of a small-diameter roller surface is much slower than that of a larger one, which can make the deposited material thicker. In other words, the effect of roller rotational speed on deposit thickness is greater than that of deposition area. The roller diameter should be selected in the range of 0.2 to 0.25 m. The production rate for a roller diameter of 0.25 m increases to 4.47 kgs−1 per meter of nozzle length compared with 3.05 kgs−1 per meter of nozzle length for a roller diameter of 0.2 m.

Fig. 7
figure 7

Effect of roller diameter on the deposit thickness

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The influence of the rotational speed of the mandrel on the deposit thickness is shown in Figure 8. A lower deposit thickness is obtained at a higher rotational speed. This is attributed to different deposition times. Moreover, the rotational speed should be selected in the range of 2.5 to 3 rps.

Fig. 8
figure 8

Effect of roller rotational speed on the deposit thickness

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The thickness profiles of the deposit for different roll gaps are illustrated in Figure 9. The larger the roll gap, the thinner the deposit. This must be due to the main deposition area of the rollers with a large roll gap deviating from the center of the spray cone compared with that with a small roll gap. Thus, the deposit thickness decreases. The roll gap in this case should be in the range of 0.004 to 0.005 m. However, the production rate for a roll gap of 0.005 m increases to 3.67 kgs−1 per meter of nozzle length compared with 3.05 kgs−1 per meter of nozzle length for a roll gap of 0.004 m.

Fig. 9
figure 9

Effect of roll gap on the deposit thickness

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Moreover, the determination of the minimum deposit thickness should also consider the removing of porosity and prior droplet boundaries in the deposited material.

3.3 Optimization of Processing Parameters

It is possible to optimize processing parameters quickly, easily and accurately by using an orthogonal design method. An appropriate simulation program and table are designed and arranged, with the simulation results shown in Table I. The preform geometry is sensitive to processing parameters. The diameter of roller D plays a significant role on the deposit thickness as compared with that by other processing parameters. Moreover, the effects of spray distance H and the rotational speed of the roller ω on deposit thickness are remarkable compared with that of the roll gap δ.

Table I The Uniform Designs L *16 (54) and the Orthogonal Table of Maximum Thickness
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4 Conclusions

During the spray-rolling process, the geometrical effect of the rotating mandrel is an important factor for droplet deposition. Both the deposition mass and redeposition mass are retained on the surface at the point of impact, and contribute to the growth of the surface at that point, which leads to a desired deposit thickness. This analytic model can be used directly to predict the corresponding drag-in angle, deposit thickness, and strip production rate, which agree well with the experimental results. In general, the rolling angle decreases with decreasing mass flux, and increasing roller diameter and rotational speed, spray distance, and roll gap. The production rate increases with increasing mass flux, roller diameter and rotational speed, spray distance, and roll gap. In this work, a method is presented to ensure a continued operation with a high production rate.