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Simulation of 3D grinding temperature field by using an improved finite difference method

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Abstract

3D grinding temperature field can be simulated by the finite difference method. When the finite difference method is used to calculate the temperature field, the boundary conditions need to be considered. Unfortunately, the difference equations for internal nodes and boundary nodes are not the same. For cuboid computing domains, there are twenty-six types of boundary nodes. This reduces the calculation efficiency to a certain extent. An improved finite difference method for the 3D grinding temperature field was proposed to solve this problem in this paper. By adding auxiliary nodes, the boundary nodes are converted into internal nodes, and the difference equations of the boundary nodes and internal nodes are unified. In this paper, the improved algorithm was used to simulate the 3D grinding temperature field. The temperature distribution characteristics of the grinding temperature field were analyzed. Then, the effects of heat source type, space step size, and convective heat transfer coefficient on the temperature field were studied. The results show that the improved algorithm can well simulate the 3D grinding temperature field. Finally, compared with the original algorithm, the calculation results were completely consistent, and the efficiency is improved by about 20%.

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Abbreviations

T :

is the temperature field (°C)

\( {T}_{i,j,k}^{(n)} \) :

is the temperature of the node (i, j, k) at time τ (τ = nΔτ) (°C)

T ′ :

is the extended temperature field (°C)

τ :

is the time (s)

Δτ :

is the time step (s)

n :

is the count of time step

λ :

is the thermal conductivity coefficient of the workpiece material (W/(m K))

ρ :

is the density of the workpiece material (kg/m3)

c :

is the specific heat capacity of the workpiece material (J/(kg K))

x, y, z :

are the coordinates of the workpiece (m)

∆x, ∆y, ∆z :

are the space steps in three coordinate directions of the workpiece respectively (m)

i, j, k :

are order numbers of node in there coordinates respectively

I, J, K :

are the maximum order numbers

Φ:

is the heat flow (W)

q :

is the heat flux (W m−2)

\( {q}_x^{+},{q}_x^{-},{q}_y^{+},{q}_y^{-},{q}_z^{+},{q}_z^{-} \) :

are the heat flux in the six faces of the calculation domain. x, y, z denote the axis directions and +, − denote the forward and reverse (W m−2)

q m :

is the average heat flux (W m−2)

M :

is a binary coefficient matrix

h :

is the convective heat transfer coefficient (W/(m2 °C))

T f :

is the fluid temperature (°C)

ε :

is the energy distribution coefficient

x c :

is the distance in x-axis direction from the lowest point of grinding zone (m)

F t :

is the tangential grinding force (N/s)

vs :

is the wheel speed (m/s)

v w :

is the feed rate (m/s)

l c :

is the length of contact arc (m)

b :

is the width of grinding wheel (m)

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Funding

This work is supported by the Key Project of National Nature Science Foundation of China [Grant No. U1508206], the Major State Basic Research Development Program of China [Grant No. 2017YFA0701200], the Science and Technology Planning Project of Shenyang [Grant No. 18006001], and the Fundamental Research Funds for the Central Universities [Grant No. N180306002].

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, NO. 3-11, Wenhua Road, Heping District, Shenyang, 110819, People’s Republic of China

    Hao Chen, Ji Zhao, Yuanxing Dai, Zixuan Wang & Tianbiao Yu

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  1. Hao Chen
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  2. Ji Zhao
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  3. Yuanxing Dai
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Correspondence to Ji Zhao or Tianbiao Yu.

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Chen, H., Zhao, J., Dai, Y. et al. Simulation of 3D grinding temperature field by using an improved finite difference method. Int J Adv Manuf Technol 108, 3871–3884 (2020). https://doi.org/10.1007/s00170-020-05513-5

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