# 带钢热连轧工作辊温度场与热凸度的数值模拟

(1. 东北大学 信息科学与工程学院，沈阳 110819；

2. 东北大学 研究院，沈阳 110819；

3. 宝钢集团有限公司 中央研究院，上海 201900)

Numerical simulation of temperature field and thermal crown

of work roll during hot strip rolling

LI Wei-gang 1,3 , LIU Xiang-hua 2 , GUO Zhao-hui 3

(1. College of Information Science and Engineering, Northeastern University, Shenyang 110819?

2. Research Institute of Science and Technology，Northeastern University，Shenyang 110819?

3.Research Institute, Baosteel Group Corporation, Shanghai 201900)

Abstract: Using one-by-one and equivalent boundary condition processing on work roll surface, the evolution rule of temperature field and thermal crown of work roll in a rolling campaign was studied, and the frequency characteristic of roll temperature field was discussed. According to the actual boundary condition of work roll in hot strip mills, an axially symmetric finite difference model for roll temperature field was established. The simulation results were compared and verified with the measured values of roll surface temperature and thermal expansion after a rolling campaign. The results indicate that the complex boundary conditions of work roll rotation can be replaced by equivalent boundary conditions in the calculation of roll thermal crown. It is found that the roll temperature field can be decomposed into a low frequency component and a high frequency one. And the former is main factor, while the latter only affects the 10mm region below roll surface, which is called shallow effect. The closer the region to the roll surface is, the larger the amplitude of roll temperature is, and the farther the region to the roll surface is, the longer the time of reaching steady state temperature is.

The roll thermal crown initially increases exponentially, and tends to become a stable value after rolling a certain number of coils.

Key words: work roll?temperature field?thermal contour?finite difference?equivalent boundary condition

。弯

。因此，研究工作辊的温度场与热变形行为对 板形控制有重要意义 [2] ，对轧辊的使用与管理具有重

GINZBURG [10]

、杜凤山等 [1] 、郭振宇等 [11] 、王连生等 [12] 和杨利坡等 [13] 都曾分别建立了工作辊温度场的差分

1 轧辊温度场与热凸度模型

1.1 轧辊传热的基本方程

? ? ? ? ? ? 2 2 1 u u u u 0 0 = ? ? = r r

u

l

， ) , ( t x q r u R r = ? ? = l

， 0 0 = ? ? = x x

u

l ，

) , ( t r f x

u

x =

? ? =d l

(2) ) , (

) 0 , , ( 0 x r u x r u = (3)

1.2 轧辊温度场的差分模型

Fig. 1 Axially symmetric difference model of work roll temperature field

ê ? é ÷ ? ? ? è ? D - + - ÷ ? ? ? è ? D + D D +

= + + + + 2 2 2 1 , 1 , 1 2

, 1 , r r u r u r r r r t α u u i k j i i k j i i i

k

j i k j i ]

[ ]

k

j

i k j i k j i k j i u u u

x

t

α u , 1 , 1 , 2

1

, 1 2 - + D D +

- + + - (4)

( ) =

÷ ?

? ? è

? D - - ÷ ?

? ? è

? D + - + + - + + +

1 , 1 1 , 1 1 ,

2 2 2 1 k j i i ri k j i i ri k j i r u r r m u r r m u e

3178 对边界面及边界角点根据热量平衡关系建立差分 式，结合轧辊内部节点的差分式(5)，得到方程组：

( ) ( ) ( ) ( ) ú ú ú ú ú ú ú ú ú ú ?

ù ê ê ê ê ê ê ê ê ê ê ? é = ú ú ú ú ú ú ú ú ú ú ? ù ê ê ê ê ê ê ê

ê ê ê ? é ú ú ú ú ú

ú ú ú ú ? ù ê ê ê ê ê ê ê ê ê ? é + - - + - - + - - + - - + - + + + - + - + - - - + - - - - - + - - k j k j

k

j N k j N k j k j k j N k j N r r r r r N N r r N N r N N r r N N r p p p p u u u u e e r m e r m r m e r m r m e r m K K r r r r r r r r r r r r , 1 , 2 , 1 , 1 , 1 1 , 2 1 , 1

1 ,

2 2 2 2 2 2 2 2 1 1 1 1 1 2 1 2 1 2 1 1 M M O O O O O O (6) 式中： 2 r t

α e r D D =

； i r

ri r e m = ； 2

x t α m x D D = ； 2 r r r i i D + = + ； 2 r r r i i D -

= - ； ( ) R

r R e K r 2 / D - = ；R 为轧辊半径； k

j i p , 的表达式根据各节点的边界条件得到。

1.3 轧辊热凸度模型

( ) ( ) (

) ( ) ? ò -= D

- + = - + =

11

0 , 0 0 r 1 2 d 1 2 N i i k

j i R k j r r u u R

r r u u R

rtc b n b n (7) 式中： k j rtc 为辊身轴向j 点k 时刻的热膨胀量；v 为泊 松比；β为热膨胀系数；u 0 为初始温度。

2 换热边界条件

2(b)所示为工作辊圆周方向的热换条件分区。

Fig. 2 Heat transfer boundary condition of circumferential

direction for work roll: (a) Work roll cooling diagram? (b) Boundary condition zones of heat transfer of roll circumferential direction

6种换热情况如下：1) Z 1，轧制时轧辊与带钢接

( ) f w def s s 3

2 q u u u h n u + - D + = ? ? -l

(8)

)

6 014 . 0 exp( 10 6 14

7 . 7 10 495 . 3 s 3

4

s u h × ′ - ′ = - (9)

b s r ln 1

def s

s def

c u = D (10)

( ) b w b u u h n

u

- = ? ? -l (11)

r

c s

b π 28 . 0 V L h R h × × D =

q l (12)

1) 档水板水冷(Z 3和Z 9)

ww

w

0.4 rcw 8 . 0 ww 023 . 0 l P Re h l ×

× = (13) cw ww

r v l

V Re = (14)

w

cw

w w rcw l r v c P =

(15)

l ww 为挡水板积水与工作辊接触弧长； λw 为水的导热系 数；c w 为冷却水的比热容；νcw 为冷却水流动粘度；ρw 为冷却水的密度。 2) 直接水冷(Z 4和Z 8)

1) 当工作辊表面温度u w ＜100℃时

600

3 5

. 185 4 6870 27

. 0 sp

19 . 0 cw1 P Q h × = (16)

2) 当工作辊表面温度u w ＞200℃时

600 3 5

. 185 4 200 10 90 . 2 cw

0.05

sp

08 . 0 6 cw2 ×

- × ′ = u B P Q h (17)

3) 当工作辊表面温度100℃≤u w ≤200℃时

cw2

cw

w cw

w cw1 w cw 200 100 100 100 200 h u u u u h u h × - - × - + × - =

(18)

；当Q ≥10000 L/(s?m ?2 )，B =1。

2.1.3 辐射换热(Z 2和Z 10)

( ) ( ) ( ) [ ]

4 4 w 0 w f 273

273 + - + + - = ? ? - ￥ ￥ u u u u h n

u

r s e l

(19) 式中：h f 为工作辊表面与周围空气之间的对流换热系 数；u ∞为工作辊周围空气的温度；εr 为工作辊的表面 温度；σ0 为Stenfan Boltzmann 常数。

2.2 等效方式

3180 i i i

q l q × = ?

=

10

1 equ π

2 (20)

2) 边界逐一处理，对图 2 中的 10 个分区的边界 条件进行逐一处理，根据轧辊旋转一周依次经历的不 同换热分区，依次施加相应的换热边界条件 q i ，藉此 将轧辊圆周方向坐标转化为时间坐标。

x u 0 = )，计算公

( )

?

-= + = - × =

11

2

2 1 1 , 2

0 r 1 N i i

i k i k x r r u R u (21)

3 实验验证

4 分析与讨论

4.1 轧辊温度场

3181

Fig. 4 Measured and calculated roll surface temperatures

Fig. 5 Measured and calculated roll thermal expansions

、图6 轧制过程中轧辊不同深度处温度的低频分量

3182 c 和 d 则分别在第 12 、22、40 卷后不再有明显的上 升趋势；④轧辊中心的温度曲线 e 和径向平均温度曲 线f 在整个过程中都保持上升趋势， 轧制60卷后仍未 能达到稳定值。

Fig. 7 Temperature distribution of work roll in different penetrations

during rolling process

Fig. 10 Change of roll thermal crown with rolling time during rolling process

5 结论

1) 考虑热轧带钢工作辊的边界条件， 建立了工作 辊温度场的径向隐式、轴向显式的差分模型，通过数 值模拟与现场实测轧辊表面温度和热膨胀量的对比表 明，轧辊表面温度计算值与实测值偏差在3 ℃以内， 轧辊热膨胀量计算值与实测值偏差在10μm 以内，吻 合较好。

2) 在轧辊温度场的数值模拟计算中， 提出两种边 界条件的处理方法：一是把轧辊表面分为10个区间、 6 种情况的分区逐一处理方法，二是按照各区弧长加 权的等效处理方法。对两种处理方法做了比较，发现 两者对接近轧辊表面处，温度计算结果差别较大；而 对轧辊径向平均温度和热凸度的计算结果差别不大， 等效处理方式可用于板形控制中的轧辊热凸度计算。

3) 采用上述两种处理方法进行模拟计算发现， 轧 辊温度可分解为以轧制 1卷带钢所经历时间为周期的

4) 轧制初期轧辊内温度变化较快， 热凸度呈指数 上升趋势，轧制一定数量的带钢后，吸收的热量与散 失的热量接近平衡，热凸度处于动态稳定状态，据此 可建立两段式轧辊热凸度在线计算模型，在一个轧制 计划初期采用指数型模型计算轧辊热凸度，后期热凸 度可采用恒定值。

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(编辑 何学锋)