Apparent Viscosity
Entrance Pressure Drop (Bagley Correction )Influence of Wall Slip
用入口压力降表征PVC 塑化程度双管毛细管流变仪
◆◆◆◆◆Apparent Viscosity for the Non-Newtonian Fluids
42)(R
Q
R L P dr r dv πηητγ=∆=
==
&Shear rate at the wall for the Newtonian Fluids :
∴For non-Newtonian fluids if we use the apparent shear rate then we can only calculate an Apparent Viscosity :
app w app γ
τ=
η&Apparent shear rate (
app ) for non-Newtonian fluids γ&L
r
P dr )r (dv ηητγ2⋅∆==−
=&4
8−=∆R Q
L
P π
ηShear stress at the wall:
∆=
τL P 2R  This is applicable for all types of fluids
apparent
w R Q
γπγ&&==
3
4γτη&=
For non-Newtonian fluids the Rabinowitch analysis is followed
-From the definition of the volumetric flow rate through a tube:
∫∫
−= → =R R R dr dr
dv r v r Q 0
202int
0 r Q  dr  v(r)2πππ-Applying the “no-slip”boundary condition (v(r)=0) and eliminating r with
the aid of                        and
ττπττd  )(0
23
3
∫
=dr dv r R Q
W
∆=L P 2R  τRabinowitch Analysis
app w
app γ
τ=
η&
∆=L P r 2r )( τAfter several manipulations we obtain the Rabinowitch equation
τ+γ
=
τ
+π=γ
W app W 3
W ln  d lnQ  d 4143ln  d lnQ  d 4143R Q
4&&∴The Real Viscosity of the polymer melt is:
飞鸟与蝉
W w γ
τ=
η&Rabinowitch Analysis
To obtain the “true”shear rate we must plot Q vs τw on logarithmic coordinates to evaluate the derivative dlnQ/dln τw for each point of the curve.¾For power-law fluids, it turns out that the slope is:
n
1
ln  d lnQ  d W =τ∴The Rabinowitch equation becomes:
n
41
n 3R Q 43W +π=γ
&Rabinowitch Analysis for the Power-Law Fluids
∆=
τL P 2R  W
+
=W
ln  d lnQ  414
3
τγγd app W &&根据上述的假设所作的推导，结果是有误差的，例如，有滑移、热效应、压缩性和出口效应等。但主要的是入口损失所引起的误差，故需校正。
Entrance Pressure Drop -Bagley Correction
∆P cap
∆P res ~0
∆P e = Entrance Pressure Drop R
L 2P W ∆=
τTherefore from eq. (1) we had:In the previous analysis we have assumed that the measured ∆P by the instrument corresponds to the pressure drop inside the capillary die, ∆P cap
Bagley Correction for ∆P entrance (入口校正)
∆P = ∆p ent + ∆p cap + ∆P exit
流体流过入口处之时，速度因从大口到小口而渐增、流线收敛，所以物料从料简经入口被挤入毛细管时，引起不同流速层之间粘性的摩擦能量耗敝(将流损失)，这是入口损失的第一个原因。
另一个原因是流体从大口流入小口时，在流动方向上产生速度梯度，引起弹性形变，这也要消耗能量。
这两项能量的损失，使得在毛细管入口处的压力降特别大。
Mechanism of Entrance Pressure Drop
Entrance Pressure Drop for HDPE and a Newtonian Fluid with the Same Viscosity
∆P =205.1
∆P = 205.3
The Entrance Pressure Drop is Determined by Shear Rate
instead of the Length-to-Diameter (L/D) Ratio.幸福花儿开
e, N 入口校正系数
N = 0.5e
+∆=
∆=
e R L P R L P W 22τBagley Correction for ∆P entrance (入口校正)
¾Unless a very long capillary is used (L/D>100), entrance pressure drop may considerably affect the accuracy of the measurements.¾The Bagley correction is used to correct for this, by assuming that we can represent this extra entrance pressure drop by an equivalent length of die, e:¾Three or four
capillaries are used and results are plotted as DP vs L/R:
文章的女儿叫什么Q 4
Q 3
Q 2Q 1
确定e 或N 的实验方法如下：保持一定流速Q ，即在一定的切变速率下，测定不同长径比的压力降DP ，以DP 对(L/D 或L/R )作图得一直线(贝格利Bayley 作图)，它在横坐标( L/D 或L/R )轴上的截距即是-e(或-N)。f4流星雨mv
实验证明，对于粘弹性流体，在剪切速率较大、温度较低、L/D 较小的情况下，入口效应不能忽略。
入口损失随物料的弹性增大、分子量增大、分子量分布加宽、切变速率增大、温度降低而增大。如使用较大长径比的毛细管时(如顺丁生胶用长径比为40以上，见图)，则入口的压力降与在毛细管中流动的压力降相比可以忽略，这时可以不进行入口校正。否则要逐点校正，此工作量极大，故应根据弹性的大小，选用合适的长径比.
Influence of Wall Slip
v slip
滑动界面的剪切速率：
1.
（Q 恒定）
2. 恒压型毛细管流变仪（∆P 恒定）
Q 1 = Q 2
乔乔 爱不分Q 1 < Q 2
< γ
.
L
R
P ηητγ2⋅∆==
&8=∆R Q
L
P π
η42R Q
L
r P πηγ=⋅∆=
&,
2. 恒压型毛细管流变仪（∆P 恒定）
)
(44R Q v R R Q πππγ+==
&Shear rate at the wall: No wall slip wall slip
14,,,+=v
w slip
v w app w R v γγγ&&&v
w slip
R v R ,3
24γππ&+⋅=
v
w slip
v b ,γ&=
14,,+=R
b
v w app w γγ&&R v R R Q v R R
Q 4)
(44γπππππγ&&+⋅=+=
=
corrected
w slip measured R
v R Q ,31
44γπ&+=intercept
slope 用入口压力降表征聚氯乙烯凝胶化程度
测量聚氯乙烯塑化程度的零长毛细管流变法
塑化程度或凝胶含量的确定
min
max max Q Q Q Q G −−=
右图为经过不同温度下的塑炼之后，聚氯乙烯(PVC)/抗冲改性剂(ACR) 制品通过零长毛细管的体积流量和凝胶含量。
Q min 塑化程度或凝胶含量最高的流量Q max 塑化程度或凝胶含量最低的流量
双管毛细管流变仪
零长毛细管
普通毛细管
模特周瑾1.由零长毛细管直接获得入口压力降
2.由普通毛细管获得粘度信息
3.
由零长毛细管获得弹性信息
Rotational Viscometer
¾Concentrated cylinders ¾Cone and Plate ¾Parallel plate
Extensional Viscometer
Mixer Viscometer
Sensor System
Coaxial Cylinder Viscometer ; Cylinder-Cylinder Viscometer;Couette Viscometer
¾Best for low viscosity systems, ηo <100 Pa.s ¾Good for relatively high shear rates ¾Not suitable for polymer melts
Types of the Coaxial Cylinder (Couette) Viscometer
内筒旋转式
外筒旋转式
单筒式
Design of CR (转速控制)  measuring Unite
Design of CS (应力控制) measuring Unite
基本公式-旋转圆桶粘度计
不同半径处扭矩是平衡的，但剪切力大小不同。
内桶和外桶表面的剪切应力分别为：
F
ω(r)
牛顿流体的粘度计算公式
内桶剪切速率与转速之间的关系：
(µ = η, 粘度)
Margules Equation
测量转速和扭矩大小及可得到Newtonian 流体的粘度