Acknowledgement
Abstract
Background
1. Characteristics of a good fuel
2. Poiseuille Equation
Methodology
1. Apparatus
  1. Ostwald Viscometer
  2. Pycnometer
2. Procedure
  1. Measurement of Relative density
  2. Measurement of Time of Fall
Results
1. Observations
2. Inference
Limitations of the analysis
1. Sources of Error
2. Other Limitations
References

Acknowledgement

I, name, take this opportunity to thank our principal, in-charge, chemistry|physics teacher and lab assistant who have provided us with proper guidance during the completion of our project.

I also thank my parents and group members for their coordination and support.

Abstract

The analysis of flow is important from a fluid mechanical as well as from the application point of view. Specifically, the analysis of the flow of fuels could be important in the design of internal combustion engines.

In this project, we derive the Poiseuille law and compare the relative viscosities of common fuels using an Ostwald Viscometer.

Our findings show that gasoline has a considerably low viscosity.

Background

Characteristics of a good fuel

A fuel is usually considered to be deemed “good” if it satisfies the following criteria $^{[1]}$

The good fuel should have a high calorific value, i.e, a large (negative) enthalpy of combustion.
A reasonable temperature of ignition is preferred. The fuel should neither ignite spontaneously or require an infeasibly high temperature for ignition.
The products of combustion should not be polluting.
The rate of combustion should be controllable.
The users should be able to trigger and ideally, stop the combustion when required.
The fuel should leave minimal byproducts behind.
It should be economically feasible to extract the fuel.
The fuel should be affordable to the common man.
It should have low moisture content.
It should be easy to handle, transport and store.

Additionally, it should be expected that an ideal fuel flows easily, i.e, has less viscosity. This is the expected behavior since a part of the energy produced is lost in providing kinetic energy to the fuel, which is something we want to minimize. Recent research supports that evidence. $^{[2]}$

Poiseuille Equation

The Poiseuille Equation relates the flow rate through a pipe with the associated pressure drop and viscosity of the fluid. It was discovered independently by Poiseuille and Hagen in the 19th century and has been successfully applied to many scenarios.

Derivation

We make the standard assumptions of incompressibility, Newtonian and laminar flow. Furthermore, we assume that the flow rate is constant, i.e, there are no unbalanced forces acting on the fluid.

The definition of viscosity gives:

$F_v = - \eta A \frac{dv}{dy}$

where $\eta$ is the coefficient of viscosity.

In this context, we consider a lamina of radius $r$ , and length $\Delta x$ :

$F_v = -\eta 2 \pi r \Delta x \frac{dv}{dr}$

The force arising from a pressure drop of $\Delta P$ :

$F_p = \pi r^2 \Delta P$

Because we assumed the fluid to be at equilibrium,

$F_p = - F_v \\ \implies \pi r^2 \Delta P = \eta 2 \pi r \Delta x \frac{dv}{dr} \\ \implies \frac{dv}{dr} = \frac{ \Delta P}{2 \eta \Delta x} r$

We cross multiply and integrate in order to solve the differential equation.

$dv = \frac{ \Delta P}{2 \eta \Delta x} r \cdot dr \\ \implies \int dv = \frac{ \Delta P}{2 \eta \Delta x} \int r \cdot dr \\ \implies v(r) = \frac{ \Delta P}{4 \eta \Delta x} r^2 + C$

To determine the constant, we use the no slip condition, which tells us that the flow at the walls is zero.

$v(R) = \frac{ \Delta P}{4 \eta \Delta x} R^2 + C = 0 \\ \implies C = -\frac{ \Delta P}{4 \eta \Delta x} R^2$

So, we have:

$v(r) = \frac{ \Delta P}{4 \eta \Delta x} (r^2 - R^2)$

The principle of continuity tells us

$\frac{dQ}{dt} = \int v \, dA$

where $\frac{dQ}{dt}$ is the volumetric flow rate.

Substituting the expressions,

$\frac{dQ}{dt} = \int^R_0 {\left ( \frac{ \Delta P}{4 \eta \Delta x} (r^2 - R^2) \right )\, d \left ( \pi r^2\right ) } \\ = \frac{\pi \Delta P}{2 \eta \Delta x} \int^R_0 {(r^3 - R^2 r)} \, dr$

Evaluating the integral above gives us the Poiseuille Equation:

$\frac{dQ}{dt} = \frac{\pi}{8} \frac{\Delta P}{\Delta x} \frac{1}{\eta}R^4$

Methodology

Apparatus

Ostwald Viscometer

The Ostwald Viscometer, named after the Russian-German chemist is one of the most common U-tube viscometers in laboratories.

One of the arms of the Ostwald viscometer consist of a bulb mounted on top of a precisely narrow capillary. When used, the fluid is drawn up to the mark in the bulb and allowed to flow. The time it takes to reach the lower mark is noted.

We rearrange Poiseuille’s formula as follows:

$\frac{Q}{t} = \frac{\pi}{8} \frac{\Delta P}{\Delta x} \frac{1}{\eta}R^4 \implies \eta = \frac{\pi}{8} \frac{\Delta P}{\Delta x} \frac{t}{Q}R^4$

It is evident that $\Delta P = h \rho g$ , i.e, the pressure difference provided by the liquid columns.

$\eta = \frac{\pi}{8} \frac{h \rho g}{\Delta x} \frac{t}{Q} R^4$

Because the difference in the height of the liquid columns remain almost same, and all the other terms cancel out,

$\frac{\eta_1}{\eta_2} = \frac{\rho_1 t_1}{\rho_2 t_2}$

Using the standard viscosity and density of water, one could measure the viscosity of liquids with this apparatus.

Pycnometer

The Pycnometer, or the specific gravity bottle is a small glass vessel with a close fitting stopper with a capillary that lets air bubbles escape.

The Pycnometer is first weighed when empty, then weighed with water and then a desired fluid.

The calculations involved is straightforward.

$\frac{\rho_{fluid}}{\rho_{water}} = \frac{W_{fluid}}{V} \times \frac{V}{W_{water}} \\ = \frac{W_{fluid}}{W_{water}} \\ = \frac{W_{fluid+bottle}-W_{bottle}}{W_{water+bottle}-W_{bottle}}$

Procedure

Measurement of Relative density

The weight of the empty bottle is measured in a digital balance and noted down as $w_1$ .
The bottle is filled with water and the weight is noted down as $w_2$ .
For each of the four fuels, the bottle is carefully washed and the weight of them is noted down as $w_i$
The relative density ( $\rho_i$ ) is calculated as $\frac{w_i - w_1}{w_2 - w_1}$

Measurement of Time of Fall

The following procedure is repeated for all the fuels:

The viscometer is washed with acetone and clamped on a stand.
20 ml of the liquid is pipetted out and put in the broader arm of the viscometer.
The liquid is drawn up to the mark on the narrow arm by suction.
Using a stopwatch, the time required for the fluid level to fall to the lower mark is measured.

Results

Observations

Fuel/Fluid	Relative Density	Time of Fall	Relative Viscosity
Water	1	92s	1
Kerosene	0.79	198s	1.70
Biodiesel	0.88	565s	5.40
Diesel	0.84	340s	3.10
Gasoline	0.73	57s	0.45

Inference

The relative velocities of the fuels are in the following order:

Biodiesel > Diesel > Kerosene ( > Water) > Gasoline

It is interesting to note that gasoline has a surprisingly low viscosity.

Limitations of the Analysis

Sources of Error

The capillary tube might not be uniformly thick.
The Poiseuille’s formula does not hold during the setup of the flow, during which most assumptions we stated in the derivation are violated.
The pressure difference is actually changing due to the flow. This is not accounted for.
Temperature changes during the measurement might alter the viscosity.

Other Limitations

Viscosity is not the only factor involved in the flow rate. There are a number of other factors, like the applied pressure and the kinematic viscosity that matters.
Other critical aspects often outweigh the importance of viscosity in the selection of an appropriate fuel.

References

Engineering Chemistry by A.K. Pahari, B.S. Chauhan
http://www.truckinginfo.com/article/story/2009/09/research-shows-oil-viscosity-affects-fuel-economy.aspx

Contents