MTX221 2023
HOMEWORK ASSIGNMENT
release: 1 (29 Sept 2023)
DUE DATE: Friday 10 November 2023 at 18:00 (6 pm)
Early submission is recommended. 10% penalization per day late, including weekends (from
18:01 to 17:59).
This is an individual assignment which will mainly be computer graded. It has been
custom-made according to your student number. Group work may NOT be submitted.
• ONLY USE THE OFFICIAL APPENDIX BOOKLET FOR THERMODYNAMIC DATA.
• Start each main Part (A and B) on a new page and work neatly.
• All question must be done fully and all calculations MUST be shown. All final answers
must be underlined clearly.
• After you completed your assignment on paper, you must type in your calculated answers
on Click-UP and also upload PDF scans of your full calculation workings to Click-UP. Your
workings must be handwritten.
• Unless indicated otherwise, only typed-in answers on Click-UP will be graded.
In this assignment, some of your input values / material types will be determined by your
student number. 𝛼1 = the first digit of your student number, 𝛼2 = the second digit and 𝛼3 =
the third digit etc.
Example:
u
2
1
1
3
0
7
9
3
For the example student number above: 𝛼1 = 2, 𝛼5 = 0 and 𝛼7 = 9.
Do not use these numbers, but use those from your student number. If your student number
starts with “04”, keep the first 0 (example 04123123).
When you scan your upload, your student card (with photo visible), must be placed onto the
first page. If you have lost your student card, then your driver’s licence, smart identity card,
passport photo page, or identity book photo page must be visible to the auditor.
There are several questions spread across parts A and B, which must be included in a single
PDF submission.
1.
Confirm your student number and enter it on Click-UP in the provided answer box.
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PART A: Fundamentals (Prof Dirker)
Refer to Figure 1. Consider a rudimentary thermodynamic assembly. It consists of a rigid
cylindrical boiler vessel, a steam tube, a turbine, and a collection sump. Steam is produced
in the boiler from liquid water and forced through the turbine to produce shaft power.
𝑑
E
D
Turbine
Boiler
B
Shaft
C
Water
A
F
Steam
𝐻
𝐷
Flames / heat source
G
Water
Collection
sump
Figure 1 Thermodynamic assembly
The boiler has an inner diameter of 𝐷, an inner height 𝐻, and a stainless-steel wall thickness
of 𝑏. Heat from a combustion process (at a flame temperature of 𝑇𝑓 = 900°C) is transferred to
liquid water on the bottom surface of the boiler via these heat transfer mechanisms:
(i) convective heat transfer on the flame side with a heat transfer coefficient, ℎ𝑏𝑜 = 20 W/m2K,
(ii) conduction heat transfer through the stainless steel with a thermal conductivity of 𝑘𝑠𝑠 =
15 W/mK, and (iii) nucleate boiling convection with a heat transfer coefficient of approximately
ℎ𝑏𝑖 = 15 kW/m2K, the rest of the boiler surfaces lose heat (𝑄̇𝑏𝑜 ) at a total rate of 800 W.
The steam tube has a diameter of 𝑑, a wall thickness of 𝑏, and a length of 𝐿, and is also
stainless-steel. Heat loss from the tube is via inner and outer surface heat transfer coefficients
of 140 W/m2K (ℎ𝑡𝑖 ) and 5 W/m2K (ℎ𝑡𝑜 ) respectively, as well as conduction. The pressure loss
in the steam tube is Δ𝑃 = 1 MPa.
The turbine is adiabatic with an isentropic efficiency of 𝜂. It is designed to operate at an inlet
pressure of 𝑃𝐷 , and an outlet pressure (atmosphere) of 𝑃𝑜 = 95 kPa. Ignore the pressure loss
in the steam tube after the turbine. E represent the state where half of the work in the turbine
was extracted from the steam turbine (IE: 𝐷 𝑤𝐸 = 𝐸 𝑤𝐹 ).
The collection sump is open to the atmosphere at 𝑇𝑜 = 25°C.
As water boils, the liquid level drops from an initial height of 𝑙1 until no more liquid is left in the
boiler. Let lines A and B in the figure always remain in the liquid and steam regions
respectively during the process. Ignore vertical pressure differences due to gravity. Use the
following inputs based on your student number digits:
Description
Boiler internal diameter
Boiler internal height
Wall thickness
Tube diameter
Tube length
Equation
𝐷 = 2+(𝛼3 /10)
𝐻 = 3+(𝛼4 /10)
𝑏 = 30+ 𝛼5
𝑑 = 60+𝛼6 × 3
𝐿 = 45+𝛼7 × 4
2
Unit
m
m
mm
mm
m
Turbine isentropic efficiency
Turbine inlet pressure
Initial liquid height
2.
𝜂 = 0.9 − (𝛼8 /100)
𝑃𝐷 = 5 + 𝛼7
𝑙1 = 2.9 − 𝛼6 × 0.2
MPa
m
Do the following calculations by hand and submit your workings as a PDF on Click-UP.
You must fill in your answers on Click-UP because you will be auto-graded.
While the water is boiling and steam is flowing through the turbine, calculate:
2.1
2.2
2.3
2.4
2.5
2.6
Pressure at points A and B (single value). (kPa)
Temperature at points A and B (single value). (°C)
The heat transfer rate into the water from the flame. (kW)
Net (total) heat transfer rate for the boiler. (kW)
Initial vapour quality inside the boiler at state 1. (%)
The rate at which water is being boiled and turned into steam Hint: Consider a
one-second time-lapse and draw a control surface around the mass of water that
turns into vapour and analyse it from first principles. (kg/s)
Assume that for any given length of time the amount of mass that turns into vapour in
the boiler is equal to the amount of mass passing point C towards the turbine. Use that
mass flow rate (where relevant) for questions 2.7 to 2.20 and calculate:
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
The increase in specific kinetic energy from B to C (J/kg)
Estimated heat transfer rate for the steam tube based on its inlet temperature.
(kW) (Use this value onwards.)
The vapour quality at point D. (%)
The specific work of the turbine if it was ideal. (kJ/kg)
The actual specific work of the turbine. (kJ/kg)
Torque produced on the shaft if it rotates at 20 Hz. (Nm)
Rate of entropy generation in the turbine. (W/K)
Pressure at point E (kPa). Hint: this requires an iterative solution approach.
Show all steps and attempts in your calculations. Skip it if you get stuck.
Minimum heat transfer rate between state F and G to fully condense the steam
back to liquid. (kW)
For the entire process (initial state until all the water has boiled away) calculate:
2.16
2.17
2.18
2.19
2.20
Mass of water that boiled away and passed through the turbine. (kg)
Length of time before all the water is boiled away in the boiler. (hours)
Total heat transfer from the flame. (MJ)
Total work produced at turbine by using a control surface around the turbine
only (MJ)
Total work produced at the turbine by using a control surface around the boiler,
steam tube and turbine. (MJ)
3. The answers in 2.19 and 2.20 are supposed to be the same, but they are not. The
assumption before question 2.7 is incorrect. Do not redo your previous questions, but do
explain why the assumption was wrong and derive an equation that will correctly represent
the mass rate that passes by point C. Fully show your derivation on paper.
4. Fill in the multiple answer options on Click-UP based on the solution of Part A (see ClickUP).
3
PART B: Application (Dr. Bock and Mr. Atofarati)
A diesel engine powered with conventional diesel fuel with heat of combustion of 40MJ/kg has
its state before compression as (70 + 3 × 𝛼4 ) kPa and (290 + 2 × 𝛼5 ) K, and it has a
maximum temperature of 2400 K. It has a bore of 0.1 m, a stroke of 0.11 m and a compression
ratio of 19:1 and is running at 2000 RPM. Each cycle takes two revolutions and the engine
has a total of six cylinders. Assume all processes to be ideal, adopt air standard conditions
and use properties from Table A.5.
5.
Do the following calculations by hand and submit your workings as a PDF. You must fill
in your answers on Click-UP because you will be auto-graded.
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
Calculate the temperature at stage 2 (before combustion) of the engine (K).
Calculate the pressure at stage 2 (before combustion) of the engine (kPa).
Calculate the cut-off ratio for the diesel cycle.
Determine the specific volume at stage 1 (m3/kg).
Determine the specific volume at stage 2 (m3/kg).
Calculate the temperature at stage 4 (after expansion) of the engine (K).
Calculate the pressure at stage 4 (after expansion) of the engine (kPa).
Determine its thermal efficiency of the cycle (%).
What type of diesel engine is considered here, ‘2’ or ‘4‘-stroke? (type 2 or 4 into
the answer box).
Considering one of the cylinders, calculate the displacement volume (L).
Calculate the amount of specific heat added to the engine during the
combustion process (kJ/kg).
Determine the mass of the air the engine intakes in each cylinder (g).
Calculate the heat added to the engine during combustion (kJ).
Then determine the mass of the fuel needed to provide the heat added in
the previous question (in grams).
Calculate the specific heat rejected from the engine during the exhaust
process (kJ/kg).
Obtain the net work done by a single cylinder of engine in a single cycle.
Calculate the overall engine power (kW).
6.
On your PDF submission provide the 𝑃 − 𝑣 and 𝑇 − 𝑠 diagram for this diesel cycle
labelling the various states and their respective properties.
7.
The diesel engine described before uses a conventional diesel. If the combustion fuel of
this engine fuel is changed to an ideal biodiesel having a heat of combustion of:
Description
Biodiesel heat
combustion:
of
Equation
𝑞ℎ = 30 000 + 𝛼6 × 1000 + 𝛼7 × 100
Unit
kJ/kg
Do the following calculations by hand and submit your workings as a PDF. You must fill
in your answers on Click-UP because you will be auto-graded:
7.1 Calculate the amount of specific heat added during combustion (𝑞ℎ ) of the engine if it
uses the biodiesel with a heat of combustion described in the table above (kJ/kg), if
the mass of biodiesel used is the same as that of the conventional diesel
7.2 Determine the thermal efficiency of the engine described above if biodiesel is utilized
in the engine, assuming the engine has the same amount of heat rejected as in
question 5.15 above (%).
4
8.
On your PDF submission discuss the comparison and contrasts between the conventional
diesel and biodiesel when utilized in this engine in terms of their thermal efficiency and
environmental impact.
9.
Fill in the multiple answer options on Click-UP based on the solution of part B (see ClickUP).
5