Determination of Coefficient of Viscosity - Lab Manual

Determination of the Coefficient of Viscosity of Water

Using Poiseuille's Flow Method

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1. AIM

To determine the coefficient of viscosity of water using Poiseuille's flow method.

2. APPARATUS USED

  • Burette with stand
  • Capillary tube of known radius and length
  • Stop watch
  • Measuring cylinder
  • Thermometer
  • Vernier caliper
  • Screw gauge
  • Water reservoir with constant head arrangement
  • Beaker
  • Clean water
  • Spirit level

3. DIAGRAM

Experimental setup for Poiseuille's flow method

4. THEORY

Poiseuille's law describes the laminar flow of a viscous fluid through a long cylindrical pipe of uniform cross-section. The flow of liquids through narrow tubes depends on:

  • The pressure difference between the ends of the tube
  • The dimensions (length and radius) of the tube
  • The viscosity of the liquid

For a liquid flowing through a narrow tube, the volume of liquid flowing per unit time (V/t) is given by Poiseuille's law:

$$\frac{V}{t} = \frac{\pi P r^4}{8 \eta L}$$

Where:

  • V = Volume of liquid flowing
  • t = Time taken
  • P = Pressure difference between the ends
  • r = Radius of the tube
  • L = Length of the tube
  • η = Coefficient of viscosity

The pressure difference P is related to the height difference h by:

$$P = \rho g h$$

Where:

  • ρ = Density of the liquid
  • g = Acceleration due to gravity
  • h = Height difference between the ends of the tube

5. FORMULA

Substituting P = ρgh in Poiseuille's equation:

$$\frac{V}{t} = \frac{\pi r^4 \rho g h}{8 \eta L}$$

Rearranging to find the coefficient of viscosity η:

$$\eta = \frac{\pi r^4 \rho g h}{8 L \times \frac{V}{t}}$$

Or simplified:

$$\eta = \frac{\pi r^4 \rho g h \times t}{8 L \times V}$$

Where:

  • η = Coefficient of viscosity (in poise or Pa·s)
  • r = Radius of the capillary tube (in cm or m)
  • ρ = Density of water at experimental temperature (in g/cm³ or kg/m³)
  • g = Acceleration due to gravity (980 cm/s² or 9.8 m/s²)
  • h = Height of water column (in cm or m)
  • L = Length of the capillary tube (in cm or m)
  • V = Volume of water collected (in cm³ or m³)
  • t = Time taken to collect volume V (in seconds)

6. PROCEDURE

  1. Setup Preparation:
    • Fix the burette on a stand and ensure it is perfectly vertical using a spirit level.
    • Mount the capillary tube at the lower end of the burette.
    • Place the measuring cylinder beneath the capillary tube to collect the water.
  2. Dimensions Measurement:
    • Measure the length (L) of the capillary tube using a vernier caliper.
    • Determine the radius (r) of the capillary tube using a screw gauge by taking measurements at multiple points and finding the average.
  3. Temperature Measurement:
    • Record the temperature of water using a thermometer to determine its density at that temperature.
  4. Experimental Steps:
    • Fill the burette with clean water.
    • Mark two points on the burette scale (e.g., 0 ml and 20 ml).
    • Measure the vertical height (h) between the water level in the burette and the end of the capillary tube.
    • Allow water to flow through the capillary tube.
    • Using a stopwatch, measure the time (t) taken for a specific volume (V) of water to flow through the capillary tube (e.g., time for water level to drop from 0 ml to 20 ml).
    • Repeat this measurement at least three times and calculate the average time.
  5. Variation of Height:
    • Repeat the above measurements for different heights (h) by adjusting the water level in the burette.
    • For each height, take at least three readings of time.

7. OBSERVATION TABLE

Dimensions of Capillary Tube:

  • Length of capillary tube (L) = _____ cm
  • Mean radius of capillary tube (r) = _____ cm
  • Temperature of water = _____ °C
  • Density of water at observed temperature (ρ) = _____ g/cm³

Table 1: Measurement of Flow Rate

S. No. Height difference (h) in cm Volume collected (V) in cm³ Time taken (t) in seconds V/t (cm³/s) η (poise)
1
2
3
4
5

8. CALCULATIONS

For each observation:

  1. Calculate the flow rate: V/t (cm³/s)
  2. Calculate the coefficient of viscosity using the formula:
    $$\eta = \frac{\pi r^4 \rho g h \times t}{8 L \times V}$$
  3. Convert the units appropriately:
    • If r and L are in cm, h in cm, V in cm³, and t in seconds, η will be in poise (g/cm·s)
    • To convert to SI units (Pa·s), multiply by 0.1
  4. Calculate the mean value of η from all observations.

Sample Calculation:

Given:

  • r = 0.015 cm
  • L = 10 cm
  • h = 30 cm
  • V = 20 cm³
  • t = 150 s
  • ρ = 0.998 g/cm³ (at 20°C)
  • g = 980 cm/s²
\begin{align} \eta &= \frac{\pi \times (0.015)^4 \times 0.998 \times 980 \times 30 \times 150}{8 \times 10 \times 20} \\ &= \frac{\pi \times 5.0625 \times 10^{-8} \times 0.998 \times 980 \times 30 \times 150}{8 \times 10 \times 20} \\ &= 0.01 \text{ poise} = 0.001 \text{ Pa·s} \end{align}

9. RESULT

The coefficient of viscosity of water at _____ °C is determined to be _____ poise or _____ Pa·s.

(Compare with the standard value at the measured temperature and calculate percentage error)

10. PRECAUTIONS

Ensure the capillary tube is clean, straight, and free from any blockages.
Make sure the burette is perfectly vertical to ensure accurate pressure difference.
Maintain constant temperature throughout the experiment.
Use clean water to avoid any impurities that might affect the flow.
Handle the capillary tube carefully as it is fragile.
Measure the radius of the capillary tube accurately, as a small error in radius measurement leads to a large error in viscosity calculation (due to r⁴ term).
Ensure there are no air bubbles in the setup.
Keep the water reservoir level constant during measurements.
Take multiple readings for better accuracy.
Ensure laminar flow conditions by maintaining appropriate flow rates.

11. VIVA VOCE QUESTIONS

Q1: What is viscosity and how is it defined?
Viscosity is the measure of a fluid's resistance to flow. It describes the internal friction of a moving fluid. A fluid with large viscosity resists motion because its molecular makeup gives it a lot of internal friction. A fluid with low viscosity flows easily because its molecular makeup results in very little friction when it is in motion.
Q2: State Poiseuille's law and explain its limitations.
Poiseuille's law states that the volume flow rate of a viscous fluid through a tube is proportional to the pressure gradient and the fourth power of the radius, and inversely proportional to the fluid viscosity and tube length. Limitations include: it applies only to laminar flow, assumes the fluid is Newtonian, tube must be straight and of uniform cross-section, and end effects are neglected.
Q3: Why does the coefficient of viscosity decrease with increase in temperature?
With increase in temperature, the kinetic energy of molecules increases, reducing the intermolecular forces. This causes molecules to move more freely with less resistance, thus decreasing the viscosity.
Q4: Why is the fourth power of radius important in Poiseuille's equation?
The fourth power of radius (r⁴) shows that the flow rate is extremely sensitive to changes in tube radius. For example, if the radius is doubled, the flow rate increases by a factor of 16. This is because flow near the center of the tube moves much faster than flow near the walls.
Q5: Differentiate between laminar and turbulent flow.
Laminar flow occurs in layers with minimal lateral mixing, and fluid particles follow smooth paths. Turbulent flow involves irregular fluctuations and mixing, with particles moving in irregular paths. Reynolds number determines the flow type, with values below 2000 typically indicating laminar flow.
Q6: What is the effect of pressure on viscosity of liquids and gases?
For liquids, viscosity is nearly independent of pressure except at very high pressures. For gases, viscosity increases with increasing pressure due to increased molecular interactions.
Q7: Why is the coefficient of viscosity for gases much lower than for liquids?
The molecules in gases are much farther apart than in liquids, resulting in fewer intermolecular interactions and thus lower resistance to flow.
Q8: What are Newtonian and non-Newtonian fluids? Give examples.
Newtonian fluids have constant viscosity regardless of applied stress (e.g., water, air). Non-Newtonian fluids have viscosity that varies with applied stress or force (e.g., ketchup, blood, cornstarch solution).
Q9: How does the radius of the capillary tube affect the experimental determination of viscosity?
Since viscosity calculation involves r⁴, even small errors in measuring the radius lead to large errors in the viscosity value. A precise measurement of radius is crucial for accurate results.
Q10: Why is it important to maintain constant temperature during this experiment?
Viscosity is highly dependent on temperature. Even small temperature variations can significantly change the viscosity value, affecting the accuracy of results.
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