ENGG 212 (W25) Worksheet # 1 (Studio Sessions Jan. 22-24)
Rules:
1. Please sign in the ‘attendance sheet’. This is separate from this worksheet and a teaching
assistant will bring it to you.
2. Complete the worksheet as a group of three or four. When you reach a checkpoint in the
worksheet, request a TA (teaching assistant) to check your work. If all the TAs are busy
helping others, please proceed to the next problem, and ask them later. Do not hesitate to
ask questions or seek clarification. Please hand-in the completed worksheet to one of the
teaching assistants at the end of the session.
3. You can form your own group of four. Each member in a group receives same mark. Best
8 worksheets will constitute 24% of your overall grade.
4. Be respectful, collaborative, and professional to help create and maintain a learning
community.
5. If you are late, please talk to the instructor to help you join a group (possibly a group which
has fewer than 4 members) and to sign in the attendance sheet if attendance is already
taken. The TA will also record the time you signed in. If you are late by more than 30
minutes, your worksheet mark will be deducted proportionately.
Suggested way to work as a team (for Cooperative learning)
1. Form a group of four.
Last Name
1
First Name
ID (last 3 digits)
2
3
4
2. Select a person in your group to be the ‘explainer’. The other members are ‘questioners.
3. The ‘explainer’ explains the question (or part of the question) and possible solution to the
‘questioners. The ‘questioners’ listen and ask questions if the explainer stops talking. The
explainer writes the solution with prompt from the questioners.
4. Let each member take the role of ‘explainer’ in turn and continue.
Question Number I [ Dimensional analysis]
You are tasked with controlling the flow rate through a pipe. You find the following equation in
a textbook that describes the flow rate through a very small constriction placed within a very
wide pipe.
𝑚̇ = 𝐶D 𝐴√𝜌(𝑃0 − 𝑃𝑐 )
The textbook states that
- 𝑚̇ is the mass flow rate through the constriction (kg/s)
- 𝜌 is fluid density (kg/m3 )
- 𝐴 is area (m2 )
- 𝑃0 is the pressure far upstream of the constriction (Pa)
- 𝑃𝑐 is the pressure at the constriction.
(a) The textbook does not provide units for 𝐶𝐷 . Using dimensional analysis, express the
dimensions of 𝐶𝐷 in terms of [𝐿]𝛼 [𝑀]𝛽 [𝑡]𝛾 [𝑇]𝛿 .
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(b) In the case of no losses, Moody (1965) recommends the following equation for calculating
the mass flow rate of wet steam (i.e. two-phase water) through the constriction.
𝑚̇ = 𝐴√
2(ℎ0 − ℎ𝑡 )
𝑣𝑡2
Variable and units are the same as in part (a), but also include the following:
- ℎ0 is the specific enthalpy upstream of the constriction
- ℎ𝑡 is the specific enthalpy at the throat
- 𝑣𝑡 is the specific volume (m3 /kg ) of the steam at the throat.
Express the dimensions of ℎ0 and ℎ𝑡 in terms of [𝐿]𝛼 [𝑀]𝛽 [𝑡]𝛾 [𝑇]𝛿 .
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(c) Specific enthalpy is often reported in the units of J/kg. Express the units in terms of SI base
units. Comment on whether these units are consistent with the results of the dimensional
analysis you performed in part (b). In your own words, describe the difference between
dimensions and units.
TA CHECK POINT
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Question Number II [ Dimensional analysis]
The Energy balance of an unsteady heated continuous stirred tank reactor (CSTR) is described
by the equation below:
𝜌 𝑉 𝐶𝑃
𝑑𝑇
= 𝑈 𝐴 (𝑇𝑚 − 𝑇) − Δ𝐻𝑅 𝑉 𝑟 − 𝑄 𝜌 𝐶𝑃 (𝑇 − 𝑇𝜊 ) … … . (1)
𝑑𝑡
Where,
ρ = density (kg/m3)
V = Reactor volume (m3)
𝐶𝑃 = Heat Capacity (kJ/kg K)
T = Outlet Temperature (K)
t = time (h)
A = Area for heat transfer (m2)
Tm = Temperature of heating fluid (K)
ΔHR = Heat of Reaction (kJ/kmol)
r
= volumetric rate of reaction (kmol/m3 h)
Q = Volumetric flow rate (m3/h)
(a) Find the dimensions of each of the additive/subtractive term in equation (1) except for the
term 𝑈 𝐴 (𝑇𝑚 − 𝑇).
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(b) Determine the dimensions of the overall heat transfer coefficient U assuming equation (1)
is a valid equation.
TA CHECK POINT
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Question Number III (Pressure)
(a) A U-tube manometer is a pressure measuring device. A vertical U-tube manometer is shown
in the sketch below that is attached to a spherical vessel filled with a gas at pressure P 1. The
manometer contains a liquid of density 𝝆. The pressure acting on the surface of the liquid at the
left arm is P1 and the pressure acting on the surface of the liquid at the right arm is P2. Consider
the liquid column in the left arm between heights 𝒉𝟏 and 𝒉𝟐 (height are measured upward from
the horizontal part of the manometer upward). If the cross-sectional area of the U-tube is A, derive
the following relationship using the hints given below:
𝑷𝟐 − 𝑷𝟏 = 𝝆𝒈(𝒉𝟏 − 𝒉𝟐 ) … … . … (𝟏)
https://commons.wikimedia.org/wiki/File:Fluid_tube_manometer_P1_lt_P2_rho_h_plain.svg
[Hints: Consider the forces acting on the liquid column in the left arm between heights 𝒉𝟏 and
𝒉𝟐 , see the figure below. When finding the force at the bottom of the column, note that the
pressure in a liquid at the same height is same. Because the column is static, the sum of these
forces should be zero. Write this equation and simplify to obtain equation (1).]
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(b) A U-tube manometer contains a fluid with a density of 825 kg/m3. The difference in height of
the two columns are 35 cm. What is the pressure difference (kPa)? What would the pressure
difference be (atm) if instead of the above fluid, the manometer contained mercury (density 13600
kg/m3) and the difference in height was still 35 cm? [ Assume 1 atm =101.325 kPa, 𝒈 =
𝟗. 𝟖 𝒎/𝒔𝟐]
Question Number IV (Mole, mole fraction, mass fraction)
(a) How many moles of N atoms are present in 34 kg of ammonia (NH 3)? Additionally, how
many nitrogen atoms are there? (Given molar mass of: N = 14, H = 1)
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(b) A vessel initially contains a non-toxic gas (see the blue dots in the figure below). But then it
gets contaminated by a toxic gas (red dots). The gas mixture is considered ‘dangerous’ if the
mole fraction of the toxic gas is greater than 0.1.
(i) Assume the vessel has 250 non-toxic gas molecules for every 30 molecules of the toxic
gas. Is the gas mixture safe or dangerous?
(ii) For every 30 toxic gas molecules, find the minimum number of non-toxic particles that
will make the gas mixture safe?
TA CHECK POINT
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Question Number # V [ Dimensionless Groups]
L
Dimensionless groups arise in the study of fluid mechanics and transport phenomena. Consider
flow through a horizontal pipe. The pressure of the fluid drops as it travels due to friction (you
will study this phenomenon in chapter 8).
A research group has measured pressure drop (Table 1) in a horizontal pipe of inner diameter 0.126
m across a pipe section of length 1.5 m for different values of average fluid velocity through the
pipe. At the experimental temperature, the density of the fluid is 0.999 g/cm3 and the viscosity is
1.138 mPa.s. Density of a fluid is the ratio of its mass to its volume, and viscosity is a measure of
its resistance to flow.
Table 1 Pressure drop across a 1.5 m length pipe section for different average velocities.
Run
1
2
3
4
5
6
average velocity (m/s)
0.3
0.9
2.7
5.1
15.3
30.6
Pressure drop (Pa)
119
813
5563
16930 115778 389429
From additional measurements, they have found that the pressure drop is a function of length of
̅ ), fluid viscosity (𝝁) and density (𝝆).
pipe section (L), pipe diameter (D), average velocity (𝒖
(a)Show that the following quantities (Re and y) are dimensionless.
̅
𝑫𝒖
𝑹𝒆𝒚𝒏𝒐𝒍𝒅𝒔 𝒏𝒖𝒎𝒃𝒆𝒓, 𝑹𝒆 = 𝝁
(𝝆 )
𝒚=
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(−𝚫𝑷/𝑳)
̅𝟐
𝝆𝒖
( 𝑫 )
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𝝁
(b) Calculate the quantity 𝝆 , known as kinematic viscosity of fluid.
𝝁
(c)Calculate the quantities 𝑫/(𝝆) and 𝑫/(𝝆𝑳)
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(d) Calculate the quantities Re and y using the expressions below for runs 4 and 5 and fill out the
table given below. In the table below, values of x and y for run 1 to 3 and 6 are already
calculated (you calculate the rest).
𝑫
̅
𝑹𝒆 = [ 𝝁 ] 𝒖
(𝝆 )
𝑫 (−𝚫𝑷)
𝒚=[ ]
̅𝟐
𝑳𝝆 𝒖
Run
L, m
D (m)
µ, mPa.s
µ, Pa.s
1
2
3
4
5
6
1.5
1.5
1.5
1.5
1.5
1.5
0.126
0.126
0.126
0.126
0.126
0.126
1.138
1.138
1.138
1.138
1.138
1.138
0.001138 0.001138 0.001138 0.001138 0.001138 0.001138
ρ (g/cm3 )
0.999
0.999
0.999
0.999
0.999
0.999
rho (kg/m3 )
average
velcocity
(m/s)
Re=x
y
999
999
999
999
999
999
0.3
0.9
2.7
33182.95 99548.86 298646.6
0.111178 0.084396 0.064165
5.1
15.3
30.6
3384661
0.03497
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(e) Plot y as a function of x (x = Re, Reynolds number) using the data in part (d) in the graph
below. The (x,y) pairs for Run 1 and Run 6 are already plotted for convenience, please plot the
remaining points. From your plot, comment if the relationship between y and x is linear or nonlinear?
0.12
0.1
y
0.08
0.06
0.04
0.02
0
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
x = Re
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4000000
(f) Now, we change variables from x and y to ln(x) and ln(y). The calculations are done and given
in the table below (please verify). Plot ln(y) as a function of ln(x) where x = Re = Reynolds number.
From your graph, is the relationship between ln(y) and ln(Re) linear or non-linear?
lnx
lny
10.4098
-2.1966
11.5084
-2.4722
12.6070
-2.7463
13.2430
-2.9053
14.3416
-3.1800
15.0348
-3.3533
ln(Re)
10.0
-2.0
12.0
14.0
16.0
-2.2
-2.4
ln(y)
-2.6
-2.8
-3.0
-3.2
-3.4
-3.6
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(g) From part (f), the research group has proposed the following linear relationship between
ln (x) and ln(y):
𝐥𝐧 𝒚 = 𝒃 𝐥𝐧(𝒙) + 𝒄 … … … (𝟏) [ 𝒙 = 𝑹𝒆]
⇒ 𝐥𝐧 𝒚 = 𝐥𝐧(𝒙𝒃 ) + 𝐥𝐧 (𝒂) [ let, 𝒄 = 𝐥𝐧 (𝒂)]
⇒ 𝐥𝐧 𝒚 = 𝐥𝐧(𝒂𝒙𝒃)
⇒ 𝒚 = 𝒂𝒙𝒃
⇒ 𝒚 = 𝒂𝒙𝒃
𝑫 𝚫𝑷
⟹𝒚=[
] = 𝒂(𝑹𝒆)𝒃 … … … (𝟐)
̅𝟐
𝑳𝝆 𝒖
Find the values of a and b using run 1 and run 2 data (In upper year, you will learn how to
consider all the data points in finding a and b). Are these constants dimensional or
dimensionless?
TA CHECK POINT
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