Experimental investigation and simulation of structure and tensile properties of Tempcore treated re

2016, Hany Khalifa, G.M. Megahed, Rawia M. Hamouda, Mohamed A. Taha, EZZ steel, Sokhna, Egypt, Faculty of Engineering, Ain-Shams University, Cairo, Egypt

Introduction

템프코어 공정에 대한 소개

Methodology

공정 조건, 관련 열 조건, 마텐자이트 형성, YS 관계 등을 나타낸 flow diagram

모델1: FDM(Finite Difference Method)으로 철근의 중심에서 표면까지의 continuous cooling curves 결정
모델2: 템퍼링 온도 데이터로 마텐자이트 분율 계산
모델3: 마텐자이트 분율과 YS의 상관관계

Experimental work

봉강 스펙: B500B, C-Mn steel bar, diameter: 10, 16mm
On-line operating parameter: water flow rate, rolling speed, rolling temperature
Recorded: tempering temperature
Calculated: quenching time
Cross-sectional sample: 금속 현미경으로 촬영

Mathematical Model

C#으로 개발

모델 1: 봉강의 cooling history를 가지고 있는 thermal model
모델 2: 마텐자이트, 코어 부피를 결정하는 metallurgical model
모델 3: 이전 모델들로 tensile propreties를 계산하는 mechanical model

Thermal model

봉강이 cooling box에 들어갈때 부터 cooling bed에서 equalizing temp가 될 때까지 전체 cooling process 동안 heat flow와 temperature distribution의 모델
FDE 모델임
axial heat conduction은 radial flow에 비해 무시할 수 있는 정도임
one dimensional, cylindrical, and transient conduction problem
model assumptions
- Uniform initial temperature, which is the rolling finishing temperature of last stand
- Radial symmetry about the bar centerline
- Independence of temperature on angular displacement
- Uniform circular cross section.

시간에 따른 temperature distribution

: 아래 non-steady conduction heat transfer Eq.의 numerical solution

\eqalign{ & {\partial \over {\partial r}}\left( {k{{\partial T} \over {\partial r}}} \right) + {k \over r}\left( {{{\partial T} \over {\partial r}}} \right) = \rho {C_p}{{\partial T} \over {\partial t}} \cr & where\,\rho = density \cr & {C_p} = specific\,heat \cr & k = conductivity \cr}

Boundary condition

At the bar cross-section center

{{\partial T} \over {\partial r}}{|_{r = 0}} = 0

At the bar surface

\eqalign{ & - k{{\partial T} \over {\partial r}}{|_{r = R}} = h\left[ {T\left( {r,t} \right) - {T_\infty }} \right] \cr & where\,{T_\infty } = temperature\,of\,the\,adjacent\,fluid \cr & h = convection\,coefficient \cr}

heat transfer h 계산
- Re<5000인 경우
  \eqalign{ & h = {{\left\{ {\left( {1.86} \right){{\left( {{\mathop{\rm Re}\nolimits} \times pr} \right)}^{0.33}} \times {k_w}} \right\}} \over {Dh}} \cr & where\,{k_w} = thermal\,conductivity \cr & Dh = hydraulic\,diameter\,calculated\,as\,\left( {Dh = {{4A} \over p}} \right) \cr & p = wetted\,perimeter \cr}
Re>5000인 경우
$h = {{\left\{ {\left( {0.023} \right){{\left( {pr} \right)}^{0.3}}{{\left( {{\mathop{\rm Re}\nolimits} } \right)}^{0.8}}} \right\}} \over {Dh}}$
Re 계산
\eqalign{ & {V_{water}} = {{Q\,per\,nozzle} \over A} \cr & A = \pi \left\{ {{{\left( {Rc} \right)}^2} - {{\left( R \right)}^2}} \right\} \cr & {V_{relative}} = \left| {{V_{water}} - {V_{bar}}} \right| \cr & where\,{V_{bar}} = bar\,linear\,velocity \cr}
water velocity 계산: 노즐당 유량을 단면적으로 나눈 값
\eqalign{ & {V_{water}} = {{Q\,per\,nozzle} \over A} \cr & A = \pi \left\{ {{{\left( {Rc} \right)}^2} - {{\left( R \right)}^2}} \right\} \cr & {V_{relative}} = \left| {{V_{water}} - {V_{bar}}} \right| \cr}

Initial conditions

The model assumes that the initial temperature is uniform (T_initial) through the diameter of the bar.

Discretization

FDE 적용

fig 4.처럼 small segment로 나눔 -> one dimensional node로 나타냄(온도가 radial dimension에만 dependent하기 때문)

Explicit form of FDE

cross section, surface, centerline 세가지 node에서 Heat balance

\eqalign{ & \sum {{E_{in}} + {E_g} = \Delta v\rho {C_p}\left( {{{{T^{t + \Delta t}} - {T^t}} \over {\Delta t}}} \right)} \cr & where\,{E_{in}} = heat\,entering\,to\,the\,node \cr & {E_g} = int ernal\,heat\,generation \cr & \Delta v = volume\,of\,the\,node \cr & \Delta t = time\,step \cr}

위 Heat balance를 바탕으로 시간 t에서 온도는

boundary node: r=R
${T^{t + \Delta t}} = {{\Delta t} \over {\rho {C_p}}}\left[ {\left( {{{2k} \over {\Delta {r^2}}} - {k \over {R\Delta r}}} \right)T_2^t + \left( {{{2h} \over {\Delta r}}} \right){T_{water\,or\,air}} + \left( {{{\rho {C_p}} \over {\Delta t}} - \left( {{{2k} \over {\Delta {r^2}}} - {k \over {R\Delta r}}} \right) - {{2h} \over {\Delta {r^2}}}} \right){T^t}} \right]$
center line node: r=0
${T^{t + \Delta t}} = {{\Delta t} \over {\rho {C_p}}}\left[ {\left( {{{4k} \over {\Delta {r^2}}}} \right)T_1^t + + \left( {{{\rho {C_p}} \over {\Delta t}} - {{4k} \over {\Delta {r^2}}}} \right){T^t}} \right]$
general node 0<r<R
${T^{t + \Delta t}} = {{\Delta t} \over {\rho {C_p}}}\left[ {\left( {{k \over {\Delta {r^2}}}} \right)\left( {T_1^t + T_2^t} \right) + \left( {{k \over {2R\Delta r}}} \right)\left( {T_1^t - T_2^t} \right) + \left( {{{\rho {C_p}} \over {\Delta t}} - {2 \over {\Delta {r^2}}}} \right){T^t}} \right]$
Where T1 = 표면 방향으로 인접한 node의 온도, T2 = 중심 방향으로 인접한 node의 온도

Stability equations

boundary node r=R
$\Delta t \le {{\rho {C_p}\Delta {r^2}} \over {4k}}$
center line node: r=0
$\Delta t \le {{\rho {C_p}\Delta {r^2}} \over {2k}}$
general node 0<r<R
$\Delta t \le {{\rho {C_p}\Delta {r^2}} \over {\left[ {\left( {{{2k} \over {\Delta {r^2}}}} \right) - \left( {{k \over {R\Delta r}}} \right) + \left( {{{2h} \over {\Delta r}}} \right)} \right]}}$