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{{Language_sel|LANG=ger|ARTIKEL=Kapillarrheometer}}
{{PSM_Infobox}}
<span style="font-size:1.2em;font-weight:bold;">Capillary rheometer</span> (Author: Prof. Dr. H.-J. Radusch)

==General principles==

Capillary rheometers are used to determine the flow behaviour of polymer melts [1].

Capillary rheometers are characterised by the fact that the fluid to be examined flows through a capillary, which can have a circular cross-section, circular ring cross-section or also a rectangular cross-section (slot). Capillary rheometers are used for both low-viscosity and high-viscosity fluids. They can work discontinuously or continuously. With low-viscosity fluids, they work according to the gravity principle, whereas with high-viscosity fluids, corresponding flow pressures must be applied.

==Types of capillary rheometers==

A distinction is made between the following capillary rheometers:
*Low-pressure capillary rheometer
:*OSTWALD type
:*UBBELOHDE type
:*CANNON-FENSKE type
*High-pressure capillary rheometer
:*discontinuous (cylinder-piston system) with variable piston force
:*discontinuous (cylinder-piston system) with variable piston speed and
:*continuous (cylinder-screw system).

==Measuring principle of a capillary rheometers==

Capillary rheometers work according to the following measuring principle. After appropriate tempering, the liquid to be characterised is conveyed from a reservoir through a corresponding capillary by means of gravity or pressure. The pressure in front of the capillary drops to the ambient pressure at the end of the capillary. This pressure gradient and the volume flowing through the capillary per unit of time are measured and the rheological [[Material Parameter | parameters]] are calculated from this. When measuring with capillary viscometers, the pressure difference Δp can be specified and the volume flow Q measured (CS principle; '''c'''ontrolled '''s'''tress), or the volume flow Q can be specified and the resulting pressure difference measured (CR principle; controlled rate).

The principle curvce of the flow velocity, the shear velocity, the shear stress and the viscosity of Newtonian and non-Newtonian fluids is shown in '''Fig. 1'''. The description of the flow processes in capillaries is based on force and mass balances, a shear stress approach and boundary conditions.

[[file:capillaryrheometer1.jpg|450px]]
{|
|- valign="top"
|width="60px"|'''Fig. 1''':
|width="600px" | Schematic representation of flow velocity, shear rate, shear stress and viscosity during the flow of a Newtonian and non-Newtonian fluids (after [2])
|}

==Rheological parameters==

Equations (1) to (4) are valid for capillaries with a circular cross-section. The shear stress dependent on the radius r is calculated by

{|
|-
|width="20px"|
|width="500px" | <math> \tau_{r}=\frac{r}{2\Delta l}\Delta p</math>
|(1)
|}

with:
{|
|-
|Δl
|width="15px" |
|Capillary length between the pressure measuring points
|-
|Δp
|
|Pressure drop across the capillary segment Δl
|}

The velocity gradient as a function of the capillary radius can be determined from the volume troughput:
{|
|-
|width="20px"|
|width="500px" | <math> \dot{\gamma_{r}}=\frac{4}{\pi r^{3}}Q</math>
|(2)
|}

The time-dependent determination of a volume flow Q = V/t through a pipe of length Δl at a pressure gradient Δp is possible by means of the relationship according to HAGEN-POISSEUILLE (equation (3)), from which the viscosity η can be determined.

{|
|-
|width="20px"|
|width="500px" | <math> \frac{dV}{dt}=\frac{\pi r^{4}\Delta p}{8 \eta \Delta l}</math>
|(3)
|}

{|
|-
|width="20px"|
|width="500px" | <math> \eta = \frac{\pi r^{4}\Delta p}{8 V \Delta l}t=\frac{\pi r^{4}}{8 \Delta l}\frac{\Delta p}{Q}</math>
|(4)
|}

For capillaries with rectangular cross-section, i.e. slot with height h and width w, h << w applies to the shear stress on the wall:

{|
|-
|width="20px"|
|width="500px" | <math> \tau_{w}=\frac{h}{2}\frac{\Delta p}{\Delta l}</math>
|(5)
|}

For the velocity gradient, it follows

{|
|-
|width="20px"|
|width="500px" | <math> \dot{\gamma_{w}}=\frac{6}{w h^{2}}Q</math>
|(6)
|}

and the viscosity corresponding to

{|
|-
|width="20px"|
|width="500px" | <math> \eta = \frac{w h^{3}}{12 \Delta l}\frac {\Delta p}{Q}</math>
|(7)
|}

'''Acknowledgements'''

The editors of the encyclopaedia would like to thank [[Radusch,_Hans-Joachim|Prof. Dr. H.-J. Radusch]], [https://www.uni-halle.de Martin Luther University Halle-Wittenberg] and [https://de.wikipedia.org/wiki/Polymer_Service_Merseburg Polymer Service GmbH Merseburg] for this guest contribution.

==See also==
*[[Melt Mass-Flow Rate | Melt mass-flow rate]]

'''References'''
{|
|-valign="top"
|[1]
|Radusch, H.-J.: Determining Process-Related Properties. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing Carl Hanser Munich (2022) 3rd. Edition, pp. 55–57, (ISBN 978-1-56990-806-8; E-Book ISBN 978-1-56990-807-5; see [[AMK-Büchersammlung|AMK-Library]] under A 22)
|-valign="top"
|[2]
|Schramm, G.: Einführung in die Rheologie und Rheometrie. Gebrüder Haake, Karlsruhe, 2. Auflage (2004)
|}

[[Category:Process-related Properties]]
[[Category:Guest Contributions]]

Capillary Rheometer - Revision history