Generality of CFD-PBM coupled model for simulations of gas-liquid bubble column

doi:10.11949/j.issn.0438-1157.20181220

Abstract

Abstract:

The generality of the CFD-PBM coupled model was studied by comparing the simulation results with experimental data under different operating pressures and liquid properties. The results show that the CFD-PBM coupled model with the modified pressure factor obtained from the internal-flow bubble breakup model can well predict the influence trend of pressure on the hydrodynamics of bubble column. The gas holdup increases significantly with increasing pressure. In addition, the simulation results for different liquid viscosity and surface tension are consistent with the experimental results. With increasing liquid viscosity, the bubble breakup rate decreases, which leads to a wider bubble size distribution, lower drag correction factor and decreased gas holdup. As the surface tension decreases, the bubble breakup rate increases, which results in smaller bubbles and higher gas holdup. The CFD-PBM coupling model has good versatility because it considers the effects of pressure, liquid viscosity and surface tension on bubble coalescence, fracture and gas-liquid interaction.

Key words: bubble column, CFD-PBM coupled model, operating pressure, physical properties, gas holdup

摘要：

通过对不同操作压力和不同液体性质气液鼓泡床的模拟值与实验数据进行对比，从而验证CFD-PBM耦合模型的通用性。结果表明，CFD-PBM耦合模型在加入了气泡破碎修正因子后，可以很好地预测压力对鼓泡床流体力学行为的影响趋势，当压力升高时，气含率显著升高。不同液体黏度和表面张力条件下CFD-PBM耦合模型的模拟结果与实验结果均吻合较好。随液体黏度增大，气泡破碎速率减小，气泡尺寸分布变宽，曳力显著下降，气含率随之降低。随表面张力减小，气泡破碎速率增大，气泡变小，气含率升高。CFD-PBM耦合模型具有很好的通用性，原因在于考虑了压力、液体黏度和表面张力对气泡聚并、破碎和气液相间作用力的影响。

关键词: 鼓泡床, CFD-PBM耦合模型, 操作压力, 物性参数, 气含率

CLC Number:

TQ 021.1

Huahai ZHANG, Tiefeng WANG. Generality of CFD-PBM coupled model for simulations of gas-liquid bubble column[J]. CIESC Journal, 2019, 70(2): 487-495.

张华海, 王铁峰. CFD-PBM耦合模型模拟气液鼓泡床的通用性研究[J]. 化工学报, 2019, 70(2): 487-495.

Figures/Tables 13

Fig.1 Schematic diagram of CFD-PBM model

Table 1 Governing equations of two-fluid model

模型	主要方程和关联式
质量守恒方程	$? ? (ρ α u) i = 0$ , i = g, l
动量守恒方程	$? ? (ρ α u u) i = - α i ? P' + ? ? α μ e f f (? u + ? u T) i + F i, j + (ρ α) i g$ , i = g, l
液相k-ε模型
k方程	$? ? (ρ l α l k l u l) i = ? ? α l μ l a m, l + (μ t, l + μ t b) / σ k ? k l + α l (G k, l - ρ l ε l)$
ε方程	$? ? (ρ l α l ε l u l) i = ? ? α l μ l a m, l + (μ t, l + μ t b) / σ ε ? ε l + α l ε l k l (C s 1 G k, l - C s 2 ρ l ε l)$
湍动能产生项液相湍动黏度	$G k, l = μ e f f, l ? u l ? ? u l + ? u l T - 23 ? ? u l μ e f f, l ? ? u l + ρ l k l$ $μ t, l = C μ (ρ l k l 2 / ε l)$
湍能修正	$μ e f f, l = μ l a m, l + μ t, l + μ t b$ , $μ t b = C μ b ρ l α g d b s u g - u l$ $k l, t = k l + k l, g$ , $ε l, t = ε l + ε l, g$ , $k l, g = 12 α g C V M u s l i p 2$ , $ε l, g = α g g u s l i p$
气相湍动黏度	$μ t, g = μ t, l ρ g / ρ l$
相间作用力
曳力	$F D = C D C D 0 ∑ i = 1 M f i α g ρ l 3 C D i 4 d b i (u g - u l) u g - u l$ $C D i = m a x 24 R e i - 1 1 + 0.15 R e i 0.687, 83 E o / (E o + 4)$ $C D / C D 0 = k b, l a r g e k b, s m a l l, k b, l a r g e = 1 / m a x (1.0,90.0 α g f b, l a r g e),$ $k b, s m a l l = 1 - α g, s m a l l 1 + 22 α g, s m a l l / E o + 0.4$
虚拟质量力	$F V M = α g ρ l C V M D D t (u g - u l), C V M = 0.25$
横向升力	$F L = - ∑ i = 1 M f i C L i α g ρ l (u g - u l) ? u l ? r$ $C L i = m i n (0.288 t a n h (0121 R e i), f (E o i')) E o i' < 3.4 f (E o i') 3.4 < E o i' < 5.3 - 0.29 E o i' > 5.3$ $f (E o i') = 0.00925 E o i' 3 - 0.0995 E o i' 2 + 1.088$
湍动扩散力	$F T D = - C T D α g ρ l k l ? α ? r$
壁面润滑力	$F W = - ∑ i = 1 M 12 f i C W i α g d b i R - r - 2 - R + r - 2 ρ l u g - u l 2$

Table 1 Governing equations of two-fluid model

模型	主要方程和关联式
质量守恒方程	$? ? (ρ α u) i = 0$ , i = g, l
动量守恒方程	$? ? (ρ α u u) i = - α i ? P' + ? ? α μ e f f (? u + ? u T) i + F i, j + (ρ α) i g$ , i = g, l
液相k-ε模型
k方程	$? ? (ρ l α l k l u l) i = ? ? α l μ l a m, l + (μ t, l + μ t b) / σ k ? k l + α l (G k, l - ρ l ε l)$
ε方程	$? ? (ρ l α l ε l u l) i = ? ? α l μ l a m, l + (μ t, l + μ t b) / σ ε ? ε l + α l ε l k l (C s 1 G k, l - C s 2 ρ l ε l)$
湍动能产生项液相湍动黏度	$G k, l = μ e f f, l ? u l ? ? u l + ? u l T - 23 ? ? u l μ e f f, l ? ? u l + ρ l k l$ $μ t, l = C μ (ρ l k l 2 / ε l)$
湍能修正	$μ e f f, l = μ l a m, l + μ t, l + μ t b$ , $μ t b = C μ b ρ l α g d b s u g - u l$ $k l, t = k l + k l, g$ , $ε l, t = ε l + ε l, g$ , $k l, g = 12 α g C V M u s l i p 2$ , $ε l, g = α g g u s l i p$
气相湍动黏度	$μ t, g = μ t, l ρ g / ρ l$
相间作用力
曳力	$F D = C D C D 0 ∑ i = 1 M f i α g ρ l 3 C D i 4 d b i (u g - u l) u g - u l$ $C D i = m a x 24 R e i - 1 1 + 0.15 R e i 0.687, 83 E o / (E o + 4)$ $C D / C D 0 = k b, l a r g e k b, s m a l l, k b, l a r g e = 1 / m a x (1.0,90.0 α g f b, l a r g e),$ $k b, s m a l l = 1 - α g, s m a l l 1 + 22 α g, s m a l l / E o + 0.4$
虚拟质量力	$F V M = α g ρ l C V M D D t (u g - u l), C V M = 0.25$
横向升力	$F L = - ∑ i = 1 M f i C L i α g ρ l (u g - u l) ? u l ? r$ $C L i = m i n (0.288 t a n h (0121 R e i), f (E o i')) E o i' < 3.4 f (E o i') 3.4 < E o i' < 5.3 - 0.29 E o i' > 5.3$ $f (E o i') = 0.00925 E o i' 3 - 0.0995 E o i' 2 + 1.088$
湍动扩散力	$F T D = - C T D α g ρ l k l ? α ? r$
壁面润滑力	$F W = - ∑ i = 1 M 12 f i C W i α g d b i R - r - 2 - R + r - 2 ρ l u g - u l 2$

Table 2 Models of bubble breakup and coalescence

模型	主要方程和关联式
由湍流涡引起的破碎
破碎速率	$b (d) = ∫ 0 0.5 b (f v d) d f v$
子气泡大小分布	$β (f v, d) = 2 b (f v d) ∫ 01 b (f v d) d f v - 1$
补充方程	$b f v d = B b f v' d$ , $B = ρ g 70 d - 2800 d 2 d < 0.018 ρ g 0.35 d ≥ 0.018$ $b f v' d = 0.923 1 - α d n ε 1 / 3 ∫ λ m i n d b P b f v' d, λ λ + d 2 λ - 11 / 3 d λ$ $P b f v d, λ = ∫ 0 ∞ P b f v d, e (λ), λ P e (e (λ)) d e (λ)$ $P e (e (λ)) = (1 / e ˉ (λ)) e x p (- e (λ) / e ˉ (λ))$ , $e ˉ (λ) = 112 π λ 3 ρ c u ˉ λ 2$ $c f, m a x = m i n (2 1 / 3 - 1), e (λ) / (π d 2 σ)$ , $f v, m i n = π λ 3 σ / (6 e (λ) d 3$ $P b f v' d, e λ = f v, m a x - f v, m i n - 1 f v, m a x - f v, m i n ≥ δ & f v, m i n < f v' < f v, m a x 0 e l s e$
由大气泡不稳定引起的破碎
破碎速率	$b 2 (d) = b (d - d c 2) m / (d - d c 2) m + d c 2 m$
子气泡大小分布	$β (f v, d) = 2 δ (0.5)$
湍流涡引起的聚并速率：c_t=?_tP_t
碰撞频率	$? t d i, d j = 14 π α g, m a x α g, m a x - α g - 1 Γ i j 2 ε 1 / 3 d i + d j 2 d i 2 / 3 + d j 2 / 3 1 / 2$ $Γ i j = l b t, i j m / (l b t, i j m + h b t, i j m)$ , $l b t, i j = l b t, i 2 + l b t, j 2$ , $l b t = 0.89 d b$ , $h b, i j = (N i + N j) 1 / 3$
聚并效率	$P t (d i, d j) = e x p - 0.75 (1 + ξ i j 2) (1 + ξ i j 3) 1 / 2 (ρ g / ρ l + γ) - 1 (1 + ξ i j) - 3 W e i j 1 / 2$
不同上升速度引起的聚并：c_u=?_uP_u
碰撞频率	$? u d i, d j = 14 π α g, m a x α g, m a x - α g - 1 2 ε 1 / 3 d i + d j 2 d i 2 / 3 + d j 2 / 3 1 / 2$
聚并效率	$P u (d i, d j) = 0.5$
大气泡尾涡引起的聚并：c_w=?_wP_w
碰撞频率	$? w (d i, d j) = 12.0 Θ d i 2 u ˉ s l i p, i$ , $? w (d i, d j) = 15.4 d i 2 u ˉ s l i p, i$ , $u ˉ s l i p, i = 0.71 g d i$ $Θ = (d j - d c / 2) 6 / (d j - d c / 2) 6 + (d c / 2) 6 f o r d j ≥ d c / 2;$ $Θ = 0 f o r d j < d c / 2 w i t h d c = 4 σ / (g Δ ρ)$
聚并效率	$P w (d i, d j) = e x p - 0.46 ρ l 1 / 2 ε 1 / 3 σ - 1 / 2 d i d j / (d i + d j) 5 / 6$

Table 2 Models of bubble breakup and coalescence

模型	主要方程和关联式
由湍流涡引起的破碎
破碎速率	$b (d) = ∫ 0 0.5 b (f v d) d f v$
子气泡大小分布	$β (f v, d) = 2 b (f v d) ∫ 01 b (f v d) d f v - 1$
补充方程	$b f v d = B b f v' d$ , $B = ρ g 70 d - 2800 d 2 d < 0.018 ρ g 0.35 d ≥ 0.018$ $b f v' d = 0.923 1 - α d n ε 1 / 3 ∫ λ m i n d b P b f v' d, λ λ + d 2 λ - 11 / 3 d λ$ $P b f v d, λ = ∫ 0 ∞ P b f v d, e (λ), λ P e (e (λ)) d e (λ)$ $P e (e (λ)) = (1 / e ˉ (λ)) e x p (- e (λ) / e ˉ (λ))$ , $e ˉ (λ) = 112 π λ 3 ρ c u ˉ λ 2$ $c f, m a x = m i n (2 1 / 3 - 1), e (λ) / (π d 2 σ)$ , $f v, m i n = π λ 3 σ / (6 e (λ) d 3$ $P b f v' d, e λ = f v, m a x - f v, m i n - 1 f v, m a x - f v, m i n ≥ δ & f v, m i n < f v' < f v, m a x 0 e l s e$
由大气泡不稳定引起的破碎
破碎速率	$b 2 (d) = b (d - d c 2) m / (d - d c 2) m + d c 2 m$
子气泡大小分布	$β (f v, d) = 2 δ (0.5)$
湍流涡引起的聚并速率：c_t=?_tP_t
碰撞频率	$? t d i, d j = 14 π α g, m a x α g, m a x - α g - 1 Γ i j 2 ε 1 / 3 d i + d j 2 d i 2 / 3 + d j 2 / 3 1 / 2$ $Γ i j = l b t, i j m / (l b t, i j m + h b t, i j m)$ , $l b t, i j = l b t, i 2 + l b t, j 2$ , $l b t = 0.89 d b$ , $h b, i j = (N i + N j) 1 / 3$
聚并效率	$P t (d i, d j) = e x p - 0.75 (1 + ξ i j 2) (1 + ξ i j 3) 1 / 2 (ρ g / ρ l + γ) - 1 (1 + ξ i j) - 3 W e i j 1 / 2$
不同上升速度引起的聚并：c_u=?_uP_u
碰撞频率	$? u d i, d j = 14 π α g, m a x α g, m a x - α g - 1 2 ε 1 / 3 d i + d j 2 d i 2 / 3 + d j 2 / 3 1 / 2$
聚并效率	$P u (d i, d j) = 0.5$
大气泡尾涡引起的聚并：c_w=?_wP_w
碰撞频率	$? w (d i, d j) = 12.0 Θ d i 2 u ˉ s l i p, i$ , $? w (d i, d j) = 15.4 d i 2 u ˉ s l i p, i$ , $u ˉ s l i p, i = 0.71 g d i$ $Θ = (d j - d c / 2) 6 / (d j - d c / 2) 6 + (d c / 2) 6 f o r d j ≥ d c / 2;$ $Θ = 0 f o r d j < d c / 2 w i t h d c = 4 σ / (g Δ ρ)$
聚并效率	$P w (d i, d j) = e x p - 0.46 ρ l 1 / 2 ε 1 / 3 σ - 1 / 2 d i d j / (d i + d j) 5 / 6$

Table 3 Properties of different liquids

液体	密度/(kg/m³)	黏度/(Pa·s)	表面张力/(mN/m)
水^[7,11,25,26]	1000	0.001	72.5
葡萄糖A^[27,28]	1340	0.17	76.0
葡萄糖B^[27]	1380	0.55	76.0
甲苯^[29]	866	0.00058	28.5

Fig.2 Comparison of simulated and experimental average gas holdup at different operating pressures[22]

Fig.3 Mechanism of pressure effect

Fig.4 Comparison of calculated and experimental average gas holdup at different liquid viscosities

Fig.5 Calculated bubble size distributions at different liquid viscosities

Fig.6 Calculated drag correction coefficient at different liquid viscosities

Fig.7 Mechanism of effect of liquid viscosity

Fig.8 Comparison of calculated and experimental average gas holdup at different liquid surface tension

Fig.9 Mechanism of effect of liquid surface tension

v_i，v_k	——气泡体积，m³
α_g	——鼓泡床内气含率
β(v,v′)	——子气泡分布
δ_j,k	——狄拉克函数
ε	——湍流耗散速率，m²/s³
$ζ 1, ζ 2$	——分别为气泡颈部流动收缩和扩张系数
λ	——湍流涡尺寸，m
λ_min	——破碎最小湍流涡尺寸，m
μ	——液体黏度，Pa·s
ρ_g，ρ_l	——分别为气体和液体密度，kg/m³
σ	——液体表面张力，mN/m

v_i，v_k	——气泡体积，m³
α_g	——鼓泡床内气含率
β(v,v′)	——子气泡分布
δ_j,k	——狄拉克函数
ε	——湍流耗散速率，m²/s³
$ζ 1, ζ 2$	——分别为气泡颈部流动收缩和扩张系数
λ	——湍流涡尺寸，m
λ_min	——破碎最小湍流涡尺寸，m
μ	——液体黏度，Pa·s
ρ_g，ρ_l	——分别为气体和液体密度，kg/m³
σ	——液体表面张力，mN/m

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