Nghiên cứu tính toán số quá trình hóa rắn của hạt nhân chất lỏng

Nghiên cứu tính toán số quá trình hóa rắn của hạt nhân chất lỏng Nghiên cứu tính toán số quá trình hóa rắn của hạt nhân chất lỏng Nghiên cứu tính toán số quá trình hóa rắn của hạt nhân chất lỏng luận văn tốt nghiệp,luận văn thạc sĩ, luận văn cao học, luận văn đại học, luận án tiến sĩ, đồ án tốt nghiệp luận văn tốt nghiệp,luận văn thạc sĩ, luận văn cao học, luận văn đại học, luận án tiến sĩ, đồ án tốt nghiệp

BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI PHẠM DUY BÍNH NGHIÊN CỨU TÍNH TỐN SỐ Q TRÌNH HĨA RẮN CỦA HẠT CHẤT LỎNG LUẬN VĂN THẠC SĨ KHOA HỌC Chuyên ngành: Kỹ thuật Cơ khí Động lực NGƯỜI HƯỚNG DẪN KHOA HỌC: TS VŨ VĂN TRƯỜNG Hà Nội - 2019 Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí LỜI CAM ĐOAN Tơi – Phạm Duy Bính - Xin cam đoan nghiên cứu riêng hướng dẫn TS Vũ Văn Trường Các số liệu kết luận văn trung thực, chép tồn văn cơng trình khác Hà Nội, 23 tháng 11 năm 2018 Tác giả Phạm Duy Bính XÁC NHẬN CỦA NGƯỜI HƯỚNG DẪN KHOA HỌC Tôi, Vũ Văn Trường, xác nhận luận văn thạc sỹ khoa học tác giả Phạm Duy Bính cơng trình nghiên cứu tác giả hướng dẫn Các kết nêu luận văn trung thực đảm bảo đầy đủ tính khoa học luận văn thạc sỹ Hà nội ngày 03 tháng 05 năm 2019 Giáo viên hướng dẫn TS VŨ VĂN TRƯỜNG Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí LỜI NĨI ĐẦU Hiện tượng hóa rắn hạt chất lỏng bề mặt lạnh xuất nhiều tự nhiên công nghiệp Hiện tượng gây hại ứng dụng sản xuất công nghiệp Việc hiểu biết q trình hóa rắn hạt chất lỏng vơ quan trọng góp phần việc phịng tránh tác hại tượng ứng dụng tượng sản xuất đời sống Vì tính cấp thiết cần đề tài nghiên cứu tượng luận văn “Nghiên cứu tính tốn số q trình hóa rắn hạt chất lỏng” đời Với tư cách tác giả luận văn này, xin gửi lời cảm ơn chân thành sâu sắc đến TS Vũ Văn Trường, người trực tiếp hướng dẫn tơi để tơi hồn thành luận văn Đồng thời xin gửi lời cảm ơn chân thành đến tập thể thầy, cô giáo Viện Cơ khí động lực, phịng Đào tạo Trường Đại học Bách Khoa Hà Nội tận tình bảo, truyền đạt kiến thức cho tôi, giúp suốt thời gian học tập làm luận văn Cuối xin gửi lời cảm ơn đến gia đình, bạn bè tơi giúp đỡ ủng hộ suốt thời gian học tập làm luận văn Hà Nội, ngày 23 tháng 11 năm 2018 Học viên Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí MỤC LỤC MỞ ĐẦU Lý chọn đề tài Mục đích nghiên cứu luận văn Tóm tắt đóng góp luận văn 10 Phương pháp nghiên cứu 10 CHƯƠNG 1: TỔNG QUAN VÀ Ý NGHĨA CỦA ĐỀ TÀI 11 1.1 Hiện tượng hóa rắn hạt chất lỏng tự nhiên cơng nghiệp 11 1.2 Tình hình nghiên cứu toán giới nước 13 1.3 Các hạt chất lỏng nghiên cứu luận văn tầm quan trọng nghiên cứu 15 CHƯƠNG 2: CÁC PHƯƠNG TRÌNH CƠ BẢN VÀ PHƯƠNG PHÁP MÔ PHỎNG 17 2.1 Các phương trình tổng quát 17 2.2 Tích phân theo thời gian 18 2.3 Lưới so le 19 2.4 Rời rạc hóa phương trình tổng qt 21 2.4.1 Rời rạc thành phần đối lưu 26 2.4.2 Rời rạc thành phần khuếch tán 27 2.4.3 Phương trình áp suất 28 2.5 Rời rạc hóa phương trình lượng 30 2.6 Điều kiện ổn định 30 2.7 Điều kiện biên 30 2.8 Phương pháp theo dấu biên 31 2.7.1 Cấu trúc mặt phân cách 32 2.7.2 Chuyển động mặt phân cách 32 2.7.3 Cấu trúc lại mặt phân cách 33 2.7.4 Làm mịn tái cấu trúc hàm thị 34 Thạc sỹ khoa học Trang Luận văn thạc sỹ Chun ngành: Kỹ thuật máy thủy khí 2.7.5 Tính tốn sức căng bề mặt lưới 34 2.7.6 Tính tốn cân nhiệt bề mặt chuyển pha 35 CHƯƠNG 3: KẾT QUẢ CỦA Q TRÌNH MƠ PHỎNG VÀ THẢO LUẬN 37 3.1 Q trình mơ hóa rắn số hạt vật liệu chuyển pha 37 3.1 Nước 37 3.1.2 Silic 40 3.1.3 Gec-ma-ni 45 3.2 Đánh giá phương pháp mô 48 3.2.1 Nước 48 3.2.2 Silic gec-ma-ni 49 3.2.3 Phương pháp theo dấu biên với hạt kép 50 3.3 Xây dựng biểu đồ pha với góc ướt khác vật liệu silic 54 3.4 Thảo luận 55 CHƯƠNG 4: KẾT LUẬN VÀ HƯỚNG PHÁT TRIỂN 57 TÀI LIỆU THAM KHẢO 58 CÁC CƠNG TRÌNH ĐÃ CƠNG BỐ 61 Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí DANH MỤC HÌNH VẼ Hình 1: Băng đá hình thành cánh tuabin gió [1] Hình 2: Hạt bán dẫn chế tạo pin lượng mặt trời [2] Hình 1: Hạt nước đóng băng bề mặt lạnh [5] 12 Hình 2: Bên trái đường biên hạt nước trước sau hóa lỏng [6], bên phải so sánh mô số silic với kết mơ satunkin [7] 13 Hình 3: Thí nghiệm hóa rắn hạt nước với góc ướt khác Huang cộng [5] 14 Hình 1: Bố trí lưới so le [12] 20 Hình 2: Lưới vận tốc theo phương ngang [12] 21 Hình 3: Lưới vận tốc theo phương thẳng đứng [12] 21 Hình 4: Lưới cấu trúc thông thường 23 Hình 5: Ký hiệu sử dụng cho lưới chuẩn so le [12] 24 Hình 6: Ký hiệu sử dụng cho lưới chuẩn so le [12] 25 Hình 7: Ký pháp sử dụng cho lưới MAC so le chuẩn [12] 29 Hình 8: Điều kiện ranh giới lưới so le [13] 31 Hình 9: Cấu trúc mặt phân cách pha [13] 33 Hình 10: Thêm nút [13] 34 Hình 11: Loại bỏ thành phần [13] 35 Hình 12: Di chuyển biên chuyển pha theo nhiệt độ 36 Hình 1: Sự phát triển trường nhiệt độ theo thời gian không thứ nguyên với ϕo=76o nước 39 Hình 2: Sự phát triển trường nhiệt độ theo thời gian khơng thứ ngun với góc ướt ϕo=124o nước 41 Hình 3: Sự phát triển trường nhiệt độ theo thời gian không thứ nguyên với góc ướt ϕo=155o nước 42 Hình 4: Sự phát triển khói lượng riêng theo thời gian không thứ nguyên với ϕo=33o silic 44 Hình 5: Sự phát triển trường nhiệt độ theo thời gian khơng thứ ngun với góc ướt ϕo=33o gec-ma-ni 47 Hình 6: So sánh mô (bên phải) thử nghiệm (trái [5]) giọt nước với ϕo=76o 48 Hình 7: So sánh mô (bên phải) thử nghiệm (trái [5]) giọt nước đóng băng với ϕo=124o 48 Hình 8: So sánh mơ (bên phải) thử nghiệm (trái [5]) giọt nước đóng băng với ϕo=155o 49 Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí Hình 9: So sánh mơ (bên phải) thí nghiệm (trái [7]) với ϕo=33o giọt silic kết tinh 50 Hình 10: So sánh mơ (phải) thí nghiệm (trái [7]) với ϕo=33o giọt gec-ma-ni kết tinh 51 Hình 11: Hạt kép biến dạng dịng chảy trượt 52 Hình 12: Nghiên cứu biên hạt kép  = 9.0 với Re = 0.8 Ca = 0.1 lưới khác 53 Hình 13: So sánh biến dạng hạt kép dòng chảy trượt tính tốn với Hua cộng [21] Thông số Re=0.25, Ca=0.125 21 = 53 Hình 14: Q trình hóa rắn hạt silic với góc ướt 33o, 45o, 60o mô số 54 Hình 15: Q trình hóa rắn hạt silic với góc ướt 75o, 90o, 105o, 120o mô số 54 Hình 16: Mối liên hệ chiều cao hóa rắn với góc ướt thay đổi chiều cao trước sau hóa rắn với góc ướt silic 55 Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí DANH MỤC CÁC BẢNG Bảng 1: Các thơng số đặc tính nước 37 Bảng 2: Một số thông số không thứ nguyên quan trọng nước 38 Bảng 3: Các thơng số đặc tính silic 43 Bảng 4: Một số thông số không thứ nguyên quan trọng silic 43 Bảng 5: Các thơng số đặc tính gec-ma-ni 45 Bảng 6: Một số thông số không thứ nguyên quan trọng gec-ma-ni 46 Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí MỞ ĐẦU Lý chọn đề tài Băng đá hình thành bề mặt cánh máy cánh dẫn (như hình nước đóng băng cánh tuabin gió) gây ảnh hưởng lớn đến khả hoạt động máy [1] Đây nguyên nhân dẫn đến tình trạng máy hoạt động rung lắc, cơng suất khơng tính tốn ban đầu, tuổi thọ máy suy giảm Ngồi tượng gây nhiều vụ tai nạn thảm khốc ngành hàng khơng Do hóa rắn hạt nước bề mặt cánh máy bay mà dịng khí chảy qua profile cánh khơng cịn tính tốn, lực nâng giảm lực cản tăng Từ thấy tầm quan trọng việc nghiên cứu hạt chất lỏng hóa rắn bề mặt lạnh để tìm giải pháp để ngăn chặn tượng Ngoài tượng tiêu cực tượng hóa rắn hạt chất lỏng ta thấy tượng cịn ứng dụng cơng nghiệp Hình thể hạt bán dẫn chế tạo pin lượng mặt trời [2] Ngành công nghiệp lượng tái tạo phát triển mạnh mẽ năm gần Nguyên nhân ảnh hưởng tiêu cực việc sử dụng mức nguồn nhiên liệu hóa thạch Các nghiên cứu hóa rắn hạt chất lỏng bề mặt lạnh quan tâm ngành công nghiệp Hiện tượng ứng dụng việc chế tạo pin mặt trời thơng qua hóa rắn hạt bán dẫn để sản xuất pin mặt trời đại trà rẻ rộng rãi Thêm ví dụ làm bụi công nghiệp nhờ tượng Các hạt nước phun từ vịi phun quện lấy hạt bụi cơng nghiệp bám vào bề mặt lạnh chuẩn bị sẵn Hóa rắn chúng gom chúng lại tái sử dụng xử lý để khơng gây ô nhiễm môi trường,… Hình 1: Băng đá hình thành cánh tuabin gió [1] Thạc sỹ khoa học Trang Luận văn thạc sỹ Chuyên ngành: Kỹ thuật máy thủy khí Hình 2: Hạt bán dẫn chế tạo pin lượng mặt trời [2] Qua tác hại tượng tự nhiên ứng dụng cơng nghiệp việc hiểu rõ cần thiết Vì luận văn “Nghiên cứu mơ hóa rắn chất lỏng bề mặt lạnh” đời nhằm đóng góp phần cho hiểu biết bạn đọc nêu giải pháp ứng dụng tượng Đây sở để phát triển nghiên cứu chuyên sâu sau Mục đích nghiên cứu luận văn Mục đính luận văn nghiên cứu mô hạt chất lỏng bề mặt lạnh theo phương pháp số để đưa mơ hình mơ sát với thực tế để hiểu q trính hóa rắn hạt chất lỏng tự nhiên hạt chất lỏng công nghiệp cách khái quát dễ hiểu Từ hình thành ý tưởng cho đề tài sau Mong muốn đề tài góp phần vào việc ứng dụng tượng đời sống sản xuất Luận văn hướng tới mục tiêu cụ thể sau: - Mở rộng phương pháp mô số trực tiếp phát triển đề tài TS Vũ Văn Trường cho q trình hóa rắn hạt chất lỏng bề mặt lạnh - Bằng tính tốn mơ số, ảnh hưởng góc phát triển chất khác góc tiếp xúc chúng bề mặt với diện thay đổi thể tích đường chập ba pha lên q trình hóa rắn hạt chất lỏng bề mặt lạnh Thạc sỹ khoa học Trang 2124 T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2119~2126 Fig Effect of f0 on the evolution of the drop during solidification with Bo = -2.5: (a) Drop profile at t = 0.12; (b) temporal evolution of the drop height; (c) solidified drop profiles (solid line) at the nearly final stage of freezing The dash lines (c) represent the initial water drop shapes Fig Effects of f0 and Bo on the solidified drop profile at the almost final stage of freezing (*) indicates the breakup case liquid part simply elongates at t = 0.12, and necking or breakup does not occur for f0 = 77°, as demonstrated in the top-left frame of Fig 7(a) The liquid-gas front is necking at this time (bottom-left frame of Fig 7(a)), and breakup occurs at a later time after increasing f0 to 90° In contrast to f0 = 77°, the freezing drop has previously broken up into drops at f0 = 124° (right frame of Fig 7(a)) The initial drop shape signifi- cantly affects the breakup process during solidification in the case of pendant drops: increasing f0 enhances breakup This case is significantly exhibited in Fig 7(b) in which the drop height suddenly shortens at f0 = 90° and 124° The frozen drop after solidification shortens with an increase in f0 given the breakup despite the increase in the height of the initial water drop while f0 increases, as displayed in Fig 7(c) This T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2119~2126 2125 In terms of solidification time, the freezing process rapidly finishes with an increase in Bo or a decrease in f0 for the nonbreakup cases However, the solidification time reduces considerably as the breakup occurs Acknowledgments This study is funded by the Vietnam National Foundation for Science and Technology Development under grant number 107.03-2017.01 The author is grateful to Prof John C Wells at Ritsumeikan University (Japan) for facilitating our computing resources Fig Effects of f0 and Bo on the solidification time ts condition can be acceptable because additional water detaches from the freezing drop when f0 increases, and thus, a minimal amount of water can be frozen Fig presents the frozen drop profiles at the nearly final stage of solidification for various Bond numbers Bo and contact angles f0 This figure indicates that the drop height decreases with an increase in Bo for the case of non-breakup, whereas a breakup reduces the solidified drop height In addition, an increase in f0 increases the drop height in the case of non-breakup By contrast, an increase in f0 reduces the drop height with the breakup Moreover, the increase in f0 enlarges the breakup region as plotted in Figs and In terms of the solidification time ts, breakup causes the freezing process to consume minimal time to finish as illustrated in Fig The increase in Bo causes the drop to be extra lateral and thus reduces time to complete solidification in the case of non-breakup The solidification time is also significantly affected by the contact angle at the plate of the initial water drop: an increase in f0 increases ts This condition is understandable because the increase in f0 reduces the solidifying front length, thus leading to additional solidified liquids By contrast, an increase in f0 results in a decrease in ts, as depicted in Fig Conclusion A direct numerical investigation has been conducted for the freezing process of a water drop that is sessile on (i.e., Bo > 0) or pendant from (i.e., Bo < 0) a cold plate The front-tracking technique combined with the interpolation technique for the no-slip boundary condition is used The numerical results show that the freezing drop can break up into drops during solidification for the case of pendant drops, depending on the Bond number and initial water drop shape (in terms of the contact angle at the plate f0) For example, the freezing drop breaks up into drops when Bo £ −2.5 for f0 = 90° The increase in the value of f0 to 124° causes the freezing drop to break up at a high Bo (i.e., at Bo £ −1.5) Thus, an increase in f0 enhances the drop breakup in the case of pendant drops For the non-breakup cases, the height of the frozen drop decreases with an increase in Bo or with a decrease in f0 References [1] Y Cao, Z Wu, Y Su and Z Xu, Aircraft flight characteristics in icing conditions, Progress in Aerospace Sciences, 74 (2015) 62-80 [2] N 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convection, International J of Multiphase Flow, 89 (2017) 331-344 [25] T V Vu, Three-phase computation of solidification in an open horizontal circular cylinder, International J of Heat and Mass Transfer, 111 (2017) 398-409 [26] C.-C Liao, Y.-W Chang, C.-A Lin and J M McDonough, Simulating flows with moving rigid boundary using immersed-boundary method, Computers & Fluids, 39 (1) (2010) 152-167 [27] Z Jin, S Jin and Z Yang, Visualization of icing process of a water droplet impinging onto a frozen cold plate under free and forced convection, J of Visualization, 16 (1) (2013) 1317 [28] L Huang, Z Liu, Y Liu, Y Gou and L Wang, Effect of contact angle on water droplet freezing process on a cold flat surface, Experimental Thermal and Fluid Science, 40 (2012) 74-80 [29] F H Harlow and J E Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, (12) (1965) 2182-2189 [30] N Al-Rawahi and G Tryggvason, Numerical simulation of dendritic solidification with convection: Two-dimensional geometry, J of Computational Physics, 180 (2) (2002) 471496 [31] A Esmaeeli and G Tryggvason, Computations of film boiling Part I: Numerical method, International J of Heat and Mass Transfer, 47 (25) (2004) 5451-5461 Truong V Vu received his B.E (2007) degree in Mechanical Engineering from Hanoi University of Science and Technology (HUST), Vietnam, and his M.E (2010) and Ph.D (2013) degrees in Integrated Science and Engineering from Ritsumeikan University, Japan He is a lecturer at the School of Transportation Engineering, HUST His current interests include multiphase and free surface flows, phase change heat transfer, and numerical methods Journal of Mechanical Science and Technology 32 (5) (2018) 2111~2117 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online) DOI 10.1007/s12206-018-0420-5 Numerical investigation of dynamic behavior of a compound drop in shear flow† Truong V Vu1,*, Luyen V Vu2, Binh D Pham1 and Quan H Luu1,* School of Transportation Engineering, Hanoi University of Science and Technology, 01 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam Mokpo National University, Chonnam 534-729, Korea (Manuscript Received October 20, 2017; Revised January 17, 2018; Accepted February 12, 2018) Abstract We present a numerical investigation of the deformation and breakup of a compound drop in shear flow The numerical method used in this study is a two-dimensional front-tracking/finite difference technique for representing the interface separating two fluids by connected elements The compound drop with the initially circular and concentric inner and outer fronts is placed at the center of a domain whose top and bottom boundaries move in the opposite direction Because of the shear rate, the compound drop deforms and can break up into drops, depending on the flow conditions based on the Reynolds number Re, the Capillary number Ca and the interfacial tension ratio s21 of the outer to inner interfaces We vary Re in the range of 0.1-3.16, Ca in the range of 0.05-0.6 and s21 in the range of 0.8-3.2 to reveal the transition from the non-breakup to breakup regimes Numerical results indicate that the compound drop breaks up into drops when there's an increase in Re or Ca or a decrease in s21 beyond the corresponding critical values We also propose a phase diagram of Ca versus Re that shows the region in which the compound drop changes from the deformation mode to the breakup mode Keywords: Compound drop; Numerical investigation; Front-tracking; Breakup; Shear flow Introduction Compound drops that contain single or multiple smaller drops inside find applications in many fields of science and technology [1-7] In comparison with simple drops (i.e., single-phase drops) [8-11], the hydrodynamics of compound drops is much more complex due to the presence of two or more interfaces with multi-fluids [12] The compound drop can be formed through either compound jet breakup [13, 14], or intense shearing in a mixer [15] For applications in, for example, lab-on-chip systems [16] or biology engineering [17], the compound drop can experience deformation and/or breakup during movements Accordingly, understanding of the compound droplet’s rheological behaviors (for example, deformation and breakup) in shear flow plays an important role not only in the fluid dynamics of the compound drop but also in its industrial applications Here, we just focus on single-core compound drops that have only one single core inside [14, 18] Concerning the numerical works related to the compound drop dynamics in shear flow, a few studies have been carried out [12, 19, 20] Hua et al [19] performed two- and threedimensional computations, based on the immersed boundary method, to study the effects of the radius ratio of the inner to * Corresponding author Tel.: +84 2438692984 E-mail address: vuvantruong.pfae@gmail.com, quan.luuhong@hust.edu.vn † Recommended by Associate Editor Hyoung-gwon Choi © KSME & Springer 2018 outer interfaces, the interfacial tension ratio of the inner to the outer, and the inner drop location on the deformation of the compound drop However, the authors have not investigated how these parameters affect the breakup of the compound drop Luo et al [20] used a front-tracking method to reveal the underlying mechanisms for the deformation of the compound drop in shear flow with the consideration of the effects of the radius ratio However, the drop breakup has not been considered in this work [20] To investigate the breakup, Chen et al [12] used a volume of fluid method to investigate the deformation and breakup behaviors of compound drops in shear flow The phase diagrams of the Capillary number Ca, dynamic viscosity ratios and radius ratio were proposed Four types of breakup modes via three mechanisms (i.e., necking, end pinching, and capillary instability) were introduced However, the effects of some other parameters such as the Reynolds number Re and the interfacial tension ratio have not been considered In addition, the non-breakup-to-breakup transition has not been investigated in the work of Chen et al [12] It is clear that the former works considered either the deformation, i.e., non-breakup, (e.g., [20]) or the breakup (e.g., [12]) of the compound drop None of the above-mentioned works considers the transition from non-breakup to breakup Thus, the effects of some parameters on the transition have not been done in the literature In addition, a phase diagram, e.g., of Re vs Ca, showing the region where both modes exist is 2112 T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2111~2117 (a) Fig Compound drop deformation in shear flow W and H are the width and height of the computational domain Other parameters are defined in the text rarely found Finishing these missing gaps is the main purpose of the present study In the present study, we present a numerical investigation of the compound drop deformation and breakup in shear flow The method used is a two-dimensional front tracking/finite difference technique [14, 21] to track the evolution and breakup of the compound drop interface This method has been widely used in multiphase and multi-fluid flows [22, 23] We examine the effects of some non-dimensional parameters, such as the Reynolds number, the Capillary number, and the ratio of the interfacial tensions of the outer to the inner interfaces, on the transition from non-breakup to breakup Importantly, basing on this phase diagram, we propose a critical parameter to recognize the non-breakup or breakup regime for a certain pair of Re and Ca This investigation is important both academically and industrially, and is still lacking in the literature Mathematical formulation and numerical parameters Fig shows our investigated problem with a concentric circular compound drop immersed in an outer fluid (denoted by 3) and at the center of the domain The inner and middle fluids are denoted by and 2, respectively Thereby, the density and viscosity of the inner, middle, and outer fluids are respectively (r1,m1), (r2,m2) and (r3,m3) The radii of the inner and outer interfaces of the compound drop are R1 and R2, respectively We assume that the fluid properties of each fluid are constant, and the gravity effect is neglected All fluids are assumed laminar, immiscible, incompressible and Newtonian We treat all fluids as one fluid with variable properties In terms of the one-fluid representation, the governing equations are given as follows: ¶ ( ru ) ¶t + Đ × r uu = -Đp + Đ × m (Đu + ĐuT ) (1) + ị sk n f d (x - x f )dS f (b) Fig (a) The density field yielded from Eq (3); (b) the variation of r along the dash-line shown in (a) r21 = 10 and r31 = Đ ×u = (2) Here, u = (u, v) is the velocity vector, p is the pressure The superscript T denotes the transpose At the interfaces, denoted by f, s is the interfacial tension coefficient Accordingly, s1 and s2 are the interfacial tension coefficients of the inner and outer interfaces, respectively k is the curvature, and nf is the unit normal vector to the interface The Dirac delta function δ(x − xf) is zero everywhere except at the interface xf These above-mentioned equations are solved by the fronttracking method [14, 21] on a staggered grid with second order accuracy in time and space The interfaces are represented by connected points that move with the velocities interpolated from the background grid The points on the inner interface is used to reconstruct an indicator function Ii that has a value of 1.0 within the interface and 0.0 other A similar indicator function Io is built from the outer interface points Thereby, the value of the fluid properties at every location in the domain is given as f = éëf1I i + f2 (1 - I i ) ùû I o + f3 I o (3) where f stands for r and m The reconstruction of the indicators and using them to specify the fluid properties is the spirit of the front-tracking method, making it profoundly different T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2111~2117 Fig Grid refinement study: compound drop profiles at t = 9.0 with Re = 0.8 and Ca = 0.1 for different grid resolutions (a) t = 0.5 Fig Comparison of the compound drop deformation in shear flow between the present calculation and Hua et al [19] The parameters are Re = 0.25, Ca = 0.125 and s21 = from other direct numerical methods for multiphase flows [24] Fig shows an example of the density field reconstructed from Eq (3) We have used this technique in our previous works for three-phase computations [25, 26] (for details see Refs [24-26]) Unlike interface capturing methods (e.g., VOF [27] and level set [28] methods), we need more treatment to handle the drop topology change, i.e., breakup To so, we first calculate the distance between the centroids of every elements on the same front The front then breaks up when the distance is less than one half of a grid space This technique has been widely used in many front-tracking computations [22] The computation domain is shown in Fig with the periodic boundary conditions on the left and right At the top and bottom, the fluid moves horizontally, with a velocity U, in the opposite directions to induce a shear rate g& = 2U H Accordingly, the dynamics of the problem is governed by the Reynolds number Re, the Capillary number Ca, the radius ratio R21, the density ratios r21 and r31, the viscosity ratios m21 and m31, and the interfacial tension ratio s21 Re = r1g& R12 , m1 Ca = m1g& R1 , s1 R21 = R2 R1 (4) 2113 (b) t = 10 Fig Compound drop evolution with the normalized pressure and velocity fields with Re = 0.1, Ca = 0.3 and s21 = The velocity is normalized by U r 21 = r2 r m m s , r31 = , m 21 = , m31 = , s 21 = r1 r1 m1 m1 s1 (5) The dimensionless time t is defined as g&t In this study, we focus on the effects of the Reynolds and Capillary numbers and the interfacial tension, and thus other parameters are kept constant, i.e., R21 = and r21 = r31 = m21 = m31 = Re and Ca are varied in the ranges of 0.1-3.16 and 0.05-0.6, respectively with s21 = 0.8-3.2 The values of these parameters correspond to compound drops of such materials as water and silicon, with diameter in the order of a few hundreds micrometer Grid study and method validation To verify the numerical method, we perform a grid refinement study with Re = 0.8, Ca = 0.1 and s21 = 1.0 Fig shows the compound drop profiles at t = (i.e., at almost steady state) for five grid resolutions (64 ´ 16, 128 ´ 32, 256 ´ 64, 512 ´ 128 and 1024 ´ 256) with a computational domain size 2114 T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2111~2117 W ´ H = 24d1 ´ 6d1, where d1 = 2R1 The result obtained from the 512 ´ 128 grid agrees very well with that obtained from Fig Formation of spurious currents around the compound drop The parameters are the same as in Fig with zero shear rate (i.e., U = 0) the 1024 ´ 256 grid, while the coarser grids yield some differences Accordingly, we use 128 grid points in the vertical direction with H/d1 = for the rest of the computations presented in this paper The number of grid points in the horizontal direction depends on the width W of the computational domain, which is varied with the flow condition To validate the numerical method applied to simulate the deformation and breakup of the compound drop, we have compared our result to that reported in Ref [19] Hua et al [19] performed the numerical calculations using the immersed boundary method proposed by Peskin [29] Our computational domain size is the same as in Ref [19] with Re = 0.25, Ca = 0.125 and s21 = Fig compares the compound drop profile at steady state from our codes with that reported in Hua et al [19], yielding complete agreement Other validations of the front-tracking method for drops and jets can be found elsewhere, e.g., [14, 21, 30] Fig Compound drop deformation (at the steady state) for Re = 0.2 (t = 10), 0.4 (t = 10), 0.8 (t = 30) and 1.6 (t = 35), and compound drop breakup for Re = 3.2 (t = 95) The other parameters are Ca = 0.1 and s21 = Fig Compound drop deformation (at the steady state) for Ca = 0.1 (t = 6), 0.2 (t = 10) and 0.3 (t = 30), and drop breakup for Ca = 0.5 (t = 100) The other parameters are Re = 0.1 and s21 = T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2111~2117 2115 Fig Compound drop deformation (at the steady state) for s21 = 3.2 (t = 10), 1.6 (t = 10) and 1.0 (t = 10), and breakup for s21 = 0.8 (t = 100) The other parameters are Re = 0.1 and Ca = 0.3 Fig 10 Phase diagram of Ca versus Re with s21 = 1.0 The dash line separates two regions: breakup and non-breakup Results and discussion Fig shows the temporal evolution of the compound drop with Re = 0.1, Ca = 0.3 and s21 = At the initial stage of deformation, i.e., at t = 0.5, the compound drop slightly deforms with nearly uniform distribution of the pressure in each fluid, as shown in Fig 5(a) As time progresses, the shear flow deforms the compound drop interfaces, leading to nonuniformity in the pressure field (Fig 5(b)) As comparison to other regions the pressure is higher at the two farthest ends within each interface This high pressure is to balance with high curvatures there Thereby, the compound drop keeps such a deformed shape at its steady state [20] For static drops, spurious currents become dominant and destroy solutions [31, 32] However, in this study, the magnitude of the spurious currents as shown in Fig is much smaller than the magnitude of mean flow velocities introduced by the shear rate during the drop deformation and breakup When the drop reaches the steady state, the spurious currents would become a little dominant, but still much smaller than the shear rate-inducing flow (Fig 5) In addition, in this study, we not study beyond the steady state of the drop When the drop becomes steady, recog- nized by its Taylor deformation factor, we stop calculations Accordingly, the issues introduced by the spurious currents can be neglected To overcome the spurious current effect, many methods have been introduced, e.g., Francois et al [31] and Choi et al [32] For the front-tracking method, to so, one can calculate the interfacial tension force by a method introduced by Shin et al [33] We have applied this method to the computations of the drop solidification [25] where spurious currents become strong, and accurate calculations of surface tension forces become important However, we not apply the method of Shin et al [33] in the present study since the effect of the spurious current is very minor Next, we consider the effects of some parameters on the deformation and transition to the breakup of the compound drop in shear flow (a) Effect of the Reynolds number Re Fig shows the effect of Re on the deformation and breakup of the compound drop with Ca = 0.1 and s21 = At Re = 0.2, the shear flow deforms the outer interface of the compound drop Increasing the Reynolds number from 0.2 to 1.6 corresponding to increasing the shear rate results in more deformation of the compound drop An interesting point here is that varying Re in the range of 0.2 to 1.6 has a minor effect on the inner interface This is understandable since the deformation of the inner interface is mostly caused by vorticial flow, within the outer interface, induced by balance between the interfacial and viscous forces [20] However, when the Reynolds number increases from 1.6 to 3.2, the shear rate causes the outer interface to break up into smaller drops [19] After breakup, the rest of the outer interface that still encapsules the inner drop retracts and forms a smaller compound drop, as shown in the last frame of Fig (b) Effect of the Capillary number Ca Fig shows the effect of Ca on the deformation and breakup of the compound drop with Re = 0.1 and s21 = At Ca = 0.1, the outer interface of the compound drop slightly deforms due 2116 T V Vu et al / Journal of Mechanical Science and Technology 32 (5) (2018) 2111~2117 to the shear rate Since the Capillary number is the ratio of the viscous force to the interfacial tension force, increasing Ca corresponds to decreasing the force holding the drop in a spherical shape Accordingly, the drop deforms more as Ca increases to a higher value (i.e., Ca = 0.2 or 0.3) At Ca = 0.5, the interfacial tension force is so low that the outer interface breaks up into smaller drops in an end-pinching mode [12] (c) Effect of the interfacial tension ratio Fig shows the effect of the interfacial tension ratio s21 on the deformation and breakup of the compound drop with Re = 0.1 and Ca = 0.3 The interfacial tension force tends to hold the drop in a spherical shape while the shear force enhances drop deformation Accordingly, at a high interfacial tension ratio, i.e., s21 = 3.2, the compound drop just slightly deforms Decreasing the value of s21 corresponding to decreasing the force induced by the interfacial tension acting on the outer front results in more deformation As a result, at a low interfacial tension ratio, i.e., s21 = 0.8, the shear force dominates over the interfacial tension force and causes the drop to break up into simple drops with a smaller compound drop at the center, as shown in the last frame of Fig (d) Phase diagram of Ca versus Re As previously discussed, the transition from non-breakup to breakup of the compound drop in shear flow is strongly affected by the Capillary and Reynolds numbers Increasing Ca or Re enhances the breakup of the compound drop outer interface This is clearly shown in Fig 10 where a phase diagram of Ca (varied in the range of 0.05-0.6) versus Re (at 0.1, 0.316, 1.0 and 3.16) with s21 = 1.0 is proposed This figure indicates that the breakup occurs at Ca ³ 0.1 for Re = 3.16 Decreasing the value of Re to 0.1 the breakup region is narrow, i.e., at Ca ³ 0.375 From the phase diagram, we propose a critical parameter Wc, based on the equation shown in Fig 10, for discriminating two regions, as follows: Wc = 107.527 Re × Ca 2.388 (6) Accordingly, Wc >1 results in breakup, while no breakup occurs for Wc < Conclusion We have presented a numerical investigation of the compound drop deformation and breakup by a two-dimensional front-tracking/finite difference method The method is verified and validated through the grid refinement study and comparison with the numerical prediction for the compound drop deformation reported in Ref [19] Various parameters including the Reynolds number Re, the Capillary number Ca and the interfacial tension ratio s21 of the outer to inner interfaces are varied to reveal their effects on the transition from the deformation to breakup regimes The numerical results show that starting from non-breakup the compound drop can break up into drops when Re increases to a value beyond 0.316 for Ca = 0.3 and s21 = 1.0, or Ca increases to a value beyond 0.15 for Re = s21 = 1.0 In addition, decreasing s21 to a value £ 0.8 (with Re = 0.1 and Ca = 0.3) results in the compound drop breakup Moreover, a phase diagram of Ca (varied in the range of 0.05 - 0.6) versus Re (varied in the range of 0.1 - 3.16) is also proposed to indicate the transition from the non-breakup to breakup regions, in which high Re and high Ca enhance the drop breakup A parameter Wc = 107.527Re×Ca2.388 is then introduced, for this phase diagram, whose value > corresponds to the breakup regime while Wc < for the non-breakup However, the present study are limited to the twodimensional calculations Thus the results would be more valuable as three-dimentional simulations are performed Acknowledgment This research is funded by Hanoi University of Science and Technology (HUST) under project number T2016-PC-023 References [1] T V Vu, H Takakura, J C Wells and T Minemoto, Production of hollow spheres of eutectic tin-lead solder through a coaxial nozzle, J of Solid Mechanics and Materials Engineering, (10) (2010) 1530-1538 [2] R H Chen, M J Kuo, S L Chiu, J Y Pu and T H Lin, Impact of a 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Science, 61 (12) (2006) 3986-3996 [31] M M Francois, S J Cummins, E D Dendy, D B Kothe, J M Sicilian and M W Williams, A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J of Computational Physics, 213 (1) (2006) 141-173 [32] S Choi, M H Cho, H G Choi and J Y Yoo, A Q2Q1 integrated finite element method with the semi-implicit consistent CSF for solving incompressible two-phase flows with surface tension effect, International J for Numerical Methods in Fluids, 81 (5) (2016) 284-308 [33] S Shin, S I Abdel-Khalik, V Daru and D Juric, Accurate representation of surface tension using the level contour reconstruction method, J of Computational Physics, 203 (2) (2005) 493-516 Truong V Vu received the B.E (2007) degree in mechanical engineering from Hanoi University of Science and Technology (HUST) in Vietnam, the M.E (2010) and Ph.D (2013) degrees in integrated science and engineering from Ritsumeikan University in Japan He is a Lecturer, School of Transportation Engineering, HUST His current interests include multiphase and free surface flows, phase change heat transfer and numerical methods Quan H Luu received the B.E (2009) degree in aeronautical engineering from Hanoi University of Science and Technology (HUST) in Vietnam, the M.E (2010) degree in fluid power engineering from HUST and the Ph.D (2013) degrees in fluids, thermal and combustion engineering from ENSMA (Ecole Nationale Supérieure de Mécanique et d'Aérotechnique) in France He is a lecturer, School of Transportation Engineering, HUST Current interests include fluid mechanics, aerodynamics, thermal combustion, CFD and fluid structure interaction International Conference of Fluid Machinery and Automation Systems - ICFMAS2018 Numerical Simulation for Solidification of Water, Molten Silicon, Molten Germanium on a Cold Plate Binh D Pham, Truong V Vu* School of Transportation Engineering, Hanoi University of Science and Technology,01 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam *Email: vuvantruong.pfae@gmail.com Abstract In this paper, we present a numerical investigation of the solidification of water, molten silicon and molten germanium drops on a cold plate The numerical method used in this study is an axisymmetric fronttracking/finite difference technique that represents the interfaces separating two phases by connected elements These elements move on a rectangular fixed grid where the Navier-Stokes and energy equations are solved by a predictor-corrector scheme for the time integration and a central difference approximation for spatial derivatives The liquid drop with an initial shape as a section of a sphere is placed on the cold plate that is the source of solidification After complete solidification, the solidified drop has a horn shape because of growth angle (ε) and volume expansion In this paper, the growth angle is assumed to be constant The wetting angle (o) is also investigated Keywords: Numerical investigation; Front-tracking; growth angle; wetting angle Introduction The solidification of water, molten silicon and molten germanium drops on a cold plate were studied a lot However, the authors mainly stop at the experiment to give shape of the drops Here, a significant work is Satunkin’s study [1] The materials used in Satunkin’s experiments [1] were the molten silicon, molten germanium, molten InSb The experiments presented in this paper described the initial liquid as part of the sphere The author presented that the shapes of the droplets after complete solidification is affected by the growth angle (ε), the wetting angle (o) of each drop Huang et al [2] also presented the solidification of a water droplet on the cold plate in the paper “Effect of contact angle on the water droplet freezing process on a cold flat surface” [2] The liquid which Huang at al [2] studied is water The authors performed the experiments with different wetting angles (o) In the paper, the authors also showed the process after complete solidification However, in this study, we only study the solidification process of liquid and not study the process further In this paper, we investigate the solidification of water, molten silicon, and molten germanium drops on the cold plate through numerical simulation to give a model that reasonably reflects the experiments The method used in the paper is front-tracking [3] (one of the well-known methods for simulations of multi-phase) to describe the process from the liquid to solid state of the drops on the cold plate The wetting angles are chosen based on the experiment to compare the numerical simulation results with the 322 experimental results Specifically here, we choose: o=76o±1o for water, o=33o±1o for silicon, o=30o±1o for germanium Each material possesses its own growth angles Here, the growth angle of water is assumed to be ε=0o, with silicon ε=12o, with germanium ε=14o Mathematical Formulation and Numerical Parameters Fig shows the initial condition of the liquid drop which forms a part of a sphere placed on a cold plate The liquid drop is completely immersed in the Fig The initial shape and computational domain of a liquid drop October 27 - 28, 2018 Hanoi, Vietnam International Conference of Fluid Machinery and Automation Systems - ICFMAS2018 computational domain with the presence of three phases: solid, liquid and gas Three interfaces on the liquid drop are solid – liquid, solid – gas, liquid - gas These interfaces create a three – phase line (TPL) Navier – Stokes and energy equations are used to solve the problem with the phase boundaries represented by the front–tracking method The properties of each phase such as density ρ, viscosity μ, thermal conductivity k, heat capacity Cp are assumed to be constant in each phase The liquid and the gas are assumed to be incompressible and immiscible, the equations are: – Bond number characterizes the ratio of gravity to - the liquid drop Navier-Stokes equation:  (  u)   uu        u  uT   t (1)    k (x  x f )n f dS    f  g  f - Energy equation: ( C p T)    C pTu    kT    q  (x  x f )dS t f (2) - Continuity equation: .u       f  (x  x f ) qdS Lh   s l  (3) u=(u,v) is the velocity vector, p is the pressure, gis the gravitational acceleration, T is the temperature, f is used to impose a non-slip condition on the solidinterfaces Delta function δ(x−xf) has a zero value at every positions except for positions xf at the interfaces  represents the surface tension coefficient, q is thermal flux at the solid-liquid interface, s - solid, l - liquid, Lh is latent heat of fusion We have the following dimensionless parameters: Pr  C pl l C (T  T )  gR2 , St  pl m c , Bo  l , kl Lh  We  lU c2 R kl2   l RCl2 0  surface tension, We – Weber number represents the relation importance of the fluid's inertia compared to its surface tension, θo – initial dimensionless temperature, ρsl, ρgl – density ratios, μsl, μgl – viscosity ratios, ksl, kgl – thermal conductivity ratios, Cpsl, Cpgl – heat capacity ratios The non-dimentional time is τ=t/τc Here, τc=ρlCplR2/kl is reference time, R=[3Vo/(4pi)] -wetting radius, Vo – initial volume of 1/3 Results and Comparions 3.1 Water The problem is simulated based on the data of the experiment paper of Huang et al [2].The water droplet has an initial volume of 56.34L During the solidification process, the cold plate was kept at -8.5oC We assume that the initial water shape is part of the sphere with o=76o Fig shows the comparison of simulation results with experimental results of Huang et al [2] Considering the shape of the water droplet after complete solidification, we can see that the simulation result is very close to the experimental result This confirms the accuracy of the simulation method used in this study Fig shows the temporal evolution of the freezing process with the temperature field The important non-dimensional parameters of water are: Pr=7.5, St=0.1, Bo= 0.18 and We=5.10-3 In this paper, we use the growth angle of ε=0o, and the wetting angle of 0=76o same as those in the experiment of Huang et al[2] Because of Bo>0, the gravity is downward, acting on the cold plate We consider the evolution of temperature over time Fig 3a (at τ=0) depicts the initial condition of the problem The water drop is part of the sphere and is placed in the domain The cold plate is placed under the drop Here the lowest temperature is shown in blue At the next stage, τ = 6.1, the temperature from the cold plate causes the water drop to freeze (Fig 3b) The solidification process develops over time g g T0  Tc   ,  sl  s ,  gl  ,  gl  , sl  s Tm  Tc l l l l kg C ps C pg k ksl  s , k gl  , C psl  , C pgl  kl kl C pl C pl Here, Pr – Prandtl number characterizes the temperature diffusion ratio, St – Stefan number represents the ratio of sensible heat to latent heat, Bo 323 Fig Comparison between simulation (right) and experiment (left[2]) of a frozen water drop October 27 - 28, 2018 Hanoi, Vietnam International Conference of Fluid Machinery and Automation Systems - ICFMAS2018 a) τ=0 a) τ=0 b) τ=165 b) τ=6.1 c) τ=329.64 Fig Time evolution of density field at different stages: (a) τ=0, (b) τ=165 and (c) τ=329.64 c) τ=12.13 drop shape is part of the sphere with o=33o Fig shows the comparison of simulation results with experimental results of Satunkin[1] Considering the shape of the silicon droplet after complete solidification, we can see that the simulation result is very close to the experimental result.This confirms the accuracy of the simulation method used in this study Fig Time evolution of temperature field at different stages: (a) τ=0, (b) τ=6.1 and (c) τ=12.13 Fig Comparison between simulation (right) and experiment (left [1]) of a crystallized silicon drop and when water is frozen the drop volume increases and grows upward, causing the upward flow from the solidification surface.As a result, at τ=12.13, the increased volume of solid grown upwards produces a horn-shaped form at the top of the frozen droplet (Fig 3c) The horn-shape appearings on the solid drop matches the experimental results [2] 3.2 Silicon The problem is simulated, based on the data of the experiment paper of Satunkin [1].The silicon droplet has an initial volume of 56.34L The cold plate was kept at 1227 oC We assume that the initial silicon 324 Fig shows the time evolution of the freezing process with the density field The important nondimensional parameters of silicon are given as: Pr=8.10-3, St=0.1, Bo= 0.45 and We=0.2 In this paper, we use the growth angle of ε=12o, and the wetting angle of 0= 33o based on the experiment of Satunkin [1] Fig shows the evolution of the solidification interface over time At τ=0 (Fig 5a), the silicon drop is shown in red Here, the red corresponds to the highest density At the next stage τ=165, the solidification process develops over time Molten silicon is frozen the drop volume increases and grows upward causing the upward flow from the surface solidification (Fig 5b) After complete solidification the drop produces a horn-shaped form at the top (Fig 5c) At τ=329.64, the horn-shape appears on the solid drop matching the experimental results October 27 - 28, 2018 Hanoi, Vietnam International Conference of Fluid Machinery and Automation Systems - ICFMAS2018 a) τ=0 Fig Comparison between simulation (right) and experiment (left [1]) of the crystallized germanium drop 3.3 Germanium The problem is simulated, based on the data of the experiment paper of Satunkin[1] The germanium droplet has an initial volume of 50.70L The cold plate was kept at 800oC Assume that the initial germanium drop shape is part of the sphere with o = 33o Fig shows the comparison of simulation result with experimental result of Satunkin[1] In shape of germanium droplet after complete solidification, we can see that the simulation result is very close to the experimental result This confirms the accuracy of the simulation method used in this study Fig shows the temporal evolution over time of a germanium drop freezing process with the temperature field During the solidification process, the cold plate was kept 800oC The important nondimensional parameters of germanium are: Pr=8.10-3, St=0.02, Bo=0.5 and We=0.2 In this paper, we use the growth angle of ε=14o, and the wetting angle of 0=33o same as those in the experiment of Satunkin [1] We consider the evolution of temperature over time Fig 7a (at τ=0) describes the initial condition of the problem The germanium drop is part of the sphere and is placed in the domain The cold plate is placed under the drop Here the lowest temperature is shown in blue.At the next stage, τ=60, the temperature from the cold plate causes the germanium drop to freeze (Fig 7b) The solidification process develops over time and when molten germanium is frozen the drop volume increases and grows upward, causing the upward flow from the solidification surface As a result, at τ=111.024, the increased volume of solid grown upwards produces a horn-shaped form at the top of the frozen droplet (Fig 7c) The horn-shape appears on the solid drop matching the experimental results[2] 325 b) τ=60 c) τ=111.024 Fig Time evolution of temperature at different stages: (a) τ=0, (b) τ=60 and (c) τ=111.024 Discussion This paper studies the solidification process of water, silicon, germanium drops on a cold plate At the end of the solidification process, we obtain a solid shape in the form of a cone However, their shapes are different on the solidification drops The solidified drops of silicon and germanium are more conical than that of water, because each liquid has a different growth angle, the larger the growth angle, the top of the liquid after complete solidification is more conical The growth angle of water, silicon and germanium are respectively o, 12o, 14o In addition, the change in volume of the liquid also causes the shape of the drops at the end of the solidification process The focus in this paper is to use the font- tracking method to simulate the solidification process of different liquids and compare the results with the experiments Although the simulation results are very close to the experiments, many problems have still been unresolved For instance how non-dimensional parameters affect the solid shape and solidification rate In addition, other external factors affecting the process of solidification such as wind speed, gas pressure, etc are not considered October 27 - 28, 2018 Hanoi, Vietnam International Conference of Fluid Machinery and Automation Systems - ICFMAS2018 Conclusions The method we use in this paper is a fronttracking method [3] to simulates the solidification of liquids with volume change The growth angles of the liquids are constant We use the parameters of water, silicon and germanium in the two papers of Huang et al [2] and Satunkin [1] The simulation results have been compared with the corresponding experimental results, showing well agreement Development (NAFOSTED) under Grant number 107.03-2017.01 References [1] [2] Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology 326 [3] G A Satunkin, “Determination of growth angles, wetting angles, interfacial tensions and capillary constant values of melts,” J Cryst Growth, vol 255, no 1, pp 170–189, Jul 2003 L Huang, Z Liu, Y Liu, Y Gou, and L Wang, “Effect of contact angle on water droplet freezing process on a cold flat surface,” Exp Therm Fluid Sci., vol 40, pp 74–80, Jul 2012 G Tryggvason et al., “A Front-Tracking Method for the Computations of Multiphase Flow,” J Comput Phys., vol 169, no 2, pp 708–759, May 2001 October 27 - 28, 2018 Hanoi, Vietnam ... hóa rắn hạt chất lỏng nghiên cứu nhiều năm gần Một nghiên cứu hóa rắn hạt chất lỏng nghiên cứu Huang cộng công bố năm 2012 [5] Các tác giả nghiên cứu xem xét thực nghiệm phân 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