A fluid tested on a tube viscometer 0.75 cm in diameter and 30 cm long exhibited a pressure dr

A fluid tested on a tube viscometer 0.75 cm in diameter and...

A fluid tested on a tube viscometer $0.75 \mathrm{~cm}$ in diameter and $30 \mathrm{~cm}$ long exhibited a pressure drop of $1200 \mathrm{~Pa}$ when the flow rate was $50 \mathrm{~cm}^3 / \mathrm{s}$. (a) Calculate the apparent viscosity and the apparent rate of shear under this condition of flow. (b) If the same fluid flowing at the rate of $100 \mathrm{~cm}^3 / \mathrm{s}$ through a viscometer tube $0.75 \mathrm{~cm}$ in diameter and $20 \mathrm{~cm}$ long exhibits a pressure drop of $1300 \mathrm{~Pa}$, calculate the flow behavior and consistency indices. Assume wall effects are negligible.

A fluid tested on a tube viscometer $0.75 \mathrm{~cm}$ in diameter and $30 \mathrm{~cm}$ long exhibited a pressure drop of $1200 \mathrm{~Pa}$ when the flow rate was $50 \mathrm{~cm}^3 / \mathrm{s}$.
(a) Calculate the apparent viscosity and the apparent rate of shear under this condition of flow.
(b) If the same fluid flowing at the rate of $100 \mathrm{~cm}^3 / \mathrm{s}$ through a viscometer tube $0.75 \mathrm{~cm}$ in diameter and $20 \mathrm{~cm}$ long exhibits a pressure drop of $1300 \mathrm{~Pa}$, calculate the flow behavior and consistency indices. Assume wall effects are negligible.

Key Concepts

Flow Behavior Index

The flow behavior index is a dimensionless parameter in the power law model that indicates how a fluid's viscosity changes with shear rate. A value of one corresponds to Newtonian behavior, while values different from one indicate shear-thinning or shear-thickening behavior.

Consistency Index

The consistency index is a parameter in the power law model that scales the fluid's viscosity with the shear rate. It provides a measure of the fluid's 'thickness' or resistance to flow under a unit shear rate and is analogous to the viscosity in Newtonian fluids.

Non-Newtonian Fluid Behavior

Non-Newtonian fluids do not have a constant viscosity but instead exhibit a viscosity that changes with the rate of shear. To describe their behavior, models such as the power law are often used, which introduce parameters like the flow behavior index and consistency index.

Viscosity

Viscosity is a measure of a fluid's resistance to flow. It quantifies the internal friction within the fluid and is critical in designing systems where fluid flow is involved. In viscometers, the apparent viscosity is determined under specific flow conditions and can vary with changes in the shear rate, particularly for non?Newtonian fluids.

Pressure Drop

Pressure drop refers to the reduction in pressure along a tube due to frictional forces, which are influenced by the fluid's viscosity and the tube geometry. In viscometry, measuring the pressure drop under known flow conditions allows for the estimation of the fluid’s rheological properties.

Tube Viscometer

A tube viscometer is an instrument used to measure the viscous properties of a fluid by analyzing the pressure drop over a known length of a tube under controlled flow conditions. It provides a practical method for determining the apparent viscosity and shear rate in fluid flow experiments.

Shear Rate

Shear rate represents the rate at which adjacent layers of fluid move relative to each other, typically expressed in reciprocal seconds. In tube viscometry, it is related to the velocity profile within the tube and the tube dimensions. Its accurate determination is crucial for characterizing the flow behavior and calculating the apparent viscosity.

Transcript

00:01 So we know that the force is equaling the viscosity times the velocity times the cross -sectional area divided by l, the length between the plates.

00:11 We know that the volumetric flow rate, q, would be equalling the difference in pressures, p.

00:20 Sub 1 minus piece of 2 over r.

00:22 We know that we poise's equation this is simply saying that here r would be equal to 8 times and the viscosity multiplied by l the length divided by pi times the radius to the fourth power so capital r is simply represents the resistance and then our lowercase r represents the radius and so we can then say that q, poiseuio's law, prasayu's law for flow in a tube, for flow in a tube would be equalling the change in pressure, p sub 2 minus piece of 1 times pi r to the fourth power, divided by 8 times the viscosity times the length.

01:23 And so we can say that's then the volumetric flow rate, q, is directly proportional to, the change in pressure.

01:36 And so for part a, we can say that the volumetric flow rate would be equal in 100 centimeters cubed per second.

01:45 And here, the pressure difference increases by a factor of 1 .5.

01:50 So you simply have to multiply this by 1 .5 because the volumetric flow rate is, again, directly proportional to the pressure difference.

01:59 And so this is giving us 150 centimeters cubed per second for part b we can say then that the volumetric flow rate is equaling 100 cubic centimeters per second and here the the new fluid is three times of a greater viscosity now here we know that this volumetric flow rate is inversely proportional to one over the viscosity and so you simply have to multiply this if the viscosity is three times greater that means that you just simply have to multiply by one third and so this is giving us 33 .3 centimeters cubed per second now here for part c we know that the volumetric fluorate is inversely proportional to one over or rather inversely proportional to the length of the tube and so the volume metric fluorite here would be 100 centimeters cubed per second.

03:04 The length of the tube is now four times is has increased by a factor of four and so the new volumetric fluorate has decreased by a factor of four and this is going to be equalling 25 .0 centimeters cubed per second.

03:19 For part d we know that the radius is 0 .100 times the original again here the volumetric fluorate is directly proportional to the radius to the radius raised to the fourth power.

03:37 And so the volumetric flow rate q would be equaling 100 centimeters cubed per second.

03:44 Again, the radius is essentially 0 .1 .00 times the original radius...

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