Chapter 1, Units and Dimensions Video Solutions, Chemical Engineering. Solutions to the Problems in Chemical Engineering

Problem 1

$98 \%$ sulphuric acid of viscosity $0.025 \mathrm{~N} \mathrm{~s} / \mathrm{m}^2$ and density $1840 \mathrm{~kg} / \mathrm{m}^3$ is pumped at $685 \mathrm{~cm}^3 / \mathrm{s}$ through a 25 mm line. Calculate the value of the Reynolds number.

Ronald Prasad

Numerade Educator

01:12

Problem 2

Compare the costs of electricity at 1 p per kWh and gas at 15 p per therm.

Mayukh Banik

Numerade Educator

09:14

Problem 3

A boiler plant raises $5.2 \mathrm{~kg} / \mathrm{s}$ of steam at $1825 \mathrm{kN} / \mathrm{m}^2$ pressure, using coal of calorific value $27.2 \mathrm{MJ} / \mathrm{kg}$. If the boiler efficiency is $75 \%$, how much coal is consumed per day? If the steam is used to generate electricity, what is the power generation in kilowatts assuming a $20 \%$ conversion efficiency of the turbines and generators?

Eric Mockensturm

Numerade Educator

Problem 4

The power required by an agitator in a tank is a function of the following four variables:
(a) diameter of impeller,
(b) number of rotations of the impeller per unit time,
(c) viscosity of liquid,
(d) density of liquid.
From a dimensional analysis, obtain a relation between the power and the four variables.
The power consumption is found, experimentally, to be proportional to the square of the speed of rotation. By what factor would the power be expected to increase if the impeller diameter were doubled?

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Problem 5

It is found experimentally that the terminal settling velocity $u_0$ of a spherical particle in a fluid is a function of the following quantities:
particle diameter, $d$; buoyant weight of particle (weight of particle - weight of displaced fluid), $W$; fluid density, $\rho$, and fluid viscosity, $\mu$.
Obtain a relationship for $u_0$ using dimensional analysis.
Stokes established, from theoretical considerations, that for small particles which settle at very low velocities, the settling velocity is independent of the density of the fluid except in so far as this affects the buoyancy. Show that the settling velocity must then be inversely proportional to the viscosity of the fluid.

Victor Salazar

Numerade Educator

Problem 6

A drop of liquid spreads over a horizontal surface. What are the factors which will influence:
(a) the rate at which the liquid spreads, and
(b) the final shape of the drop?
Obtain dimensionless groups involving the physical variables in the two cases.

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02:22

Problem 7

Liquid is flowing at a volumetric flowrate of Q per unit width down a vertical surface. Obtain from dimensional analysis the form of the relationship between flowrate and film thickness. If the flow is streamline, show that the volumetric flowrate is directly proportional to the density of the liquid.

Amany Waheeb

Numerade Educator

Problem 8

Obtain, by dimensional analysis, a functional relationship for the heat transfer coefficient for forced convection at the inner wall of an annulus through which a cooling liquid is flowing.

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Problem 9

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection may be neglected in comparison with those of forced convection.

It is found by experiment that, when the flow is turbulent, increasing the flowrate by a factor of 2 always results in a $50 \%$ increase in the coefficient. How would a $50 \%$ increase in density of the fluid be expected to affect the coefficient, all other variables remaining constant?

Rashmi Sinha

Numerade Educator

07:00

Problem 10

A stream of droplets of liquid is formed rapidly at an orifice submerged in a second, immiscible liquid. What physical properties would be expected to influence the mean size of droplet formed? Using dimensional analysis obtain a functional relation between the variables.

Khoobchandra Agrawal

Numerade Educator

03:15

Problem 11

Liquid flows under steady-state conditions along an open channel of fixed inclination to the horizontal. On what factors will the depth of liquid in the channel depend? Obtain a relationship between the variables using dimensional analysis.

Chai Santi

Numerade Educator

01:55

Problem 12

Liquid flows down an inclined surface as a film. On what variables will the thickness of the liquid film depend? Obtain the relevant dimensionless groups. It may be assumed that the surface is sufficiently wide for edge effects to be negligible.

Chai Santi

Numerade Educator

01:28

Problem 13

A glass particle settles under the action of gravity in a liquid. Upon what variables would the terminal velocity of the particle be expected to depend? Obtain a relevant dimensionless grouping of the variables. The falling velocity is found to be proportional to the square of the particle diameter when other variables are kept constant. What will be the effect of doubling the viscosity of the liquid? What does this suggest regarding the nature of the flow?

James Kiss

Numerade Educator

Problem 14

Heat is transferred from condensing steam to a vertical surface and the resistance to heat transfer is attributable to the thermal resistance of the condensate layer on the surface.

What variables are expected to affect the film thickness at a point?
Obtain the relevant dimensionless groups.
For streamline flow it is found that the film thickness is proportional to the one third power of the volumetric flowrate per unit width. Show that the heat transfer coefficient is expected to be inversely proportional to the one third power of viscosity.

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02:11

Problem 15

A spherical particle settles in a liquid contained in a narrow vessel. Upon what variables would you expect the falling velocity of the particle to depend? Obtain the relevant dimensionless groups.

For particles of a given density settling in a vessel of large diameter, the settling velocity is found to be inversely proportional to the viscosity of the liquid. How would this depend on particle size?

Kudakwashe Mapiki

Numerade Educator

01:28

Problem 16

A liquid is in steady state flow in an open trough of rectangular cross-section inclined at an angle $\theta$ to the horizontal. On what variables would you expect the mass flow per unit time to depend? Obtain the dimensionless groups which are applicable to this problem.

Chai Santi

Numerade Educator

02:49

Problem 17

The resistance force on a spherical particle settling in a fluid is given by Stokes' Law. Obtain an expression for the terminal falling velocity of the particle. It is convenient to express experimental results in the form of a dimensionless group which may be plotted against a Reynolds group with respect to the particle. Suggest a suitable form for this dimensionless group.

Force on particle from Stokes' Law $=3 \pi \mu d u$; where $\mu$ is the fluid viscosity, $d$ is the particle diameter and $u$ is the velocity of the particle relative to the fluid.
What will be the terminal falling velocity of a particle of diameter $10 \mu \mathrm{~m}$ and of density $1600 \mathrm{~kg} / \mathrm{m}^3$ settling in a liquid of density $1000 \mathrm{~kg} / \mathrm{m}^3$ and of viscosity $0.001 \mathrm{Ns} / \mathrm{m}^2$ ?

If Stokes' Law applies for particle Reynolds numbers up to 0.2 , what is the diameter of the largest particle whose behaviour is governed by Stokes' Law for this solid and liquid?

Mayukh Banik

Numerade Educator

Problem 18

A sphere, initially at a constant temperature, is immersed in a liquid whose temperature is maintained constant. The time $t$ taken for the temperature of the centre of the sphere to reach a given temperature $\theta_c$ is a function of the following variables:

Diameter of sphere, $d$
Thermal conductivity of sphere, $k$
Density of sphere, $\rho$
Specific heat capacity of sphere, $C_p$
Temperature of fluid in which it is immersed, $\theta_s$.
Obtain relevant dimensionless groups for this problem.

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07:24

Problem 19

Upon what variables would the rate of filtration of a suspension of fine solid particles be expected to depend? Consider the flow through unit area of filter medium and express the variables in the form of dimensionless groups.

It is found that the filtration rate is doubled if the pressure difference is doubled. What would be the effect of raising the temperature of filtration from 293 to 313 K ?
The viscosity of the liquid is given by:

$$
\mu=\mu_0(1-0.015(T-273))
$$

where $\mu$ is the viscosity at a temperature $T \mathrm{~K}$ and $\mu_0$ is the viscosity at 273 K .

Mahnoor Amin

Numerade Educator