The volumetric efficiency of the engine, we need to determine the actual air mass flow rate (m_actual) and compare it to the theoretical air mass flow rate (m_theoretical) that would be achieved if the engine operated under ideal conditions.
Engine displacement (Vd) = 2000 cc = 0.002 m³
Mass flow rate (m) = 0.0401 kg/s
Pressure (P) = 1.5 bar = 150 kPa
Temperature (T) = 42°C = 42 + 273.15 = 315.15 K
calculate the density of air(ρ) using ideal gas law:
ρ = P / (R * T)
R is the specific gas constant for air (287.1 J/(kg·K)).
ρ = 150 kPa/(287.1J/(kg·K)*315.15 K)≈ 1.591kg/m³
Next, we can calculate the theoretical air mass flow rate using the equation:
m_theoretical = ρ * Vd * N / (60 * A)
where N is engine speed in revolutions per minute, A is cross-sectional area of the intake port.
Since the engine speed (N) is not provided in the question, we cannot calculate the theoretical mass flow rate and therefore cannot directly determine the volumetric efficiency.
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During the production of parts in a factory, it was noticed that the part had a 0.03 probability of failure. Determine the probability of only 2 failure parts being found in a sample of 100 parts. (Use Poissons).
The Poisson distribution is used to model the probability of a specific number of events occurring in a fixed time or space, given the average rate of occurrence per unit of time or space.
For instance, during the production of parts in a factory, it was noticed that the part had a 0.03 probability of failure.
The probability of only 2 failure parts being found in a sample of 100 parts can be calculated using Poisson's distribution as follows:
[tex]Mean (λ) = np = 100 × 0.03 = 3[/tex]
We know that [tex]P(x = 2) = [(λ^x) * e^-λ] / x![/tex]
Therefore, [tex]P(x = 2) = [(3^2) * e^-3] / 2! = 0.224[/tex]
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state the assumption made for deriving the efficiency
of gas turbine?
A gas turbine is a type of internal combustion engine that converts the energy of pressurized gas or fluid into mechanical energy, which can then be used to generate power. The following are the assumptions made for deriving the efficiency of a gas turbine:
Assumptions made for deriving the efficiency of gas turbine- A gas turbine cycle is made up of the following: intake, compression, combustion, and exhaust.
To calculate the efficiency of a gas turbine, the following assumptions are made: It's a steady-flow process. Gas turbine cycle air has an ideal gas behaviour. Each of the four processes is reversible and adiabatic; the combustion process is isobaric, while the other three are isentropic. Processes that occur within the combustion chamber are ideal. Inlet and exit kinetic energies of gases are negligible.
There is no pressure drop across any device. A gas turbine has no external heat transfer, and no heat is lost to the surroundings. The efficiencies of all the devices are known. The gas turbine cycle has no friction losses.
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3: Consider the Rankine power cycle using vapor and liquid. a: Draw the T-s property diagram of the Carnot Vapor Cycle and explain what kind of problem the Carnot Vapor Cycle has in each process explicitly. b: Draw a T-s diagram for the ideal Rankine cycle and label each process explicitly. c: There is an ideal reheating Rankine cycle in order to improve the efficiency of the ideal Rankine cycle. Explain the cycle and its necessity in terms of the property diagram explicitly. d: Name the gas power cycle which uses the same four processes in its cycle and draw the T-s diagram and the P-v diagram for the gas power cycle.
The T-s and P-v diagrams of different power cycles help illustrate the energy transformations that occur during each phase of the cycle.
These include the Carnot, Rankine, reheat Rankine, and gas power cycles. While the Carnot cycle is theoretically the most efficient, practical limitations reduce its applicability in real-world systems.
A T-s diagram for a Carnot cycle includes two isotherms and two adiabatic, but the low-temperature heat rejection phase can be problematic because it requires a condenser operating at unrealistically low pressures. The Rankine cycle, on the other hand, is a practical improvement over the Carnot cycle, as it allows for more feasible operating pressures. To further enhance efficiency, the reheat Rankine cycle includes an additional phase where steam is reheated before expanding further, minimizing moisture at the turbine outlet. The Brayton cycle, typically employed in gas power cycles, also involves the same four processes and can be illustrated with T-s and P-v diagrams.
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(b) Distinguish between "open loop control" and "closed loop control". (4 marks) (c) Discuss the reasons that "flexibility is necessary for manufacturing process. (4 marks) Hilla hitro (d) Discu
A safe work environment enhances the company's image and reputation, reduces the likelihood of lawsuits, and improves stakeholder relationships.
(b) Open Loop ControlOpen-loop control is a technique in which the control output is not connected to the input for sensing.
As a result, the input signal cannot be compared to the output signal, and the output is not adjusted in response to changes in the input.Closed Loop Control
In a closed-loop control system, the output signal is compared to the input signal.
The feedback loop provides input data to the controller, allowing it to adjust its output in response to any deviations between the input and output signals.
(c) Reasons for Flexibility in Manufacturing ProcessesThe following are some reasons why flexibility is essential in manufacturing processes:
New technologies and advances in technology occur regularly, and businesses must change how they operate to keep up with these trends.The need to offer new products necessitates a change in production processes.
New items must be launched to replace outdated ones or to capture new markets.
As a result, manufacturing firms must have the flexibility to transition from one product to another quickly.Effective manufacturing firms must be able to respond to alterations in the supply chain, such as an unexpected rise in demand or the unavailability of a necessary raw material, to remain competitive.
A flexible manufacturing system also allows for the adjustment of the production line to match the level of demand and customer preferences, reducing waste and increasing efficiency.(d) Discuss the Importance of Maintaining a Safe Workplace
A secure workplace can result in a variety of benefits, including increased morale and productivity among workers. The following are the reasons why maintaining a safe workplace is important:Employees' lives and well-being are protected, reducing the incidence of injuries and fatalities in the workplace.
The costs associated with occupational injuries and illnesses, such as medical treatment, workers' compensation, lost productivity, and legal costs, are reduced.
A safe work environment fosters teamwork and increases morale, resulting in greater job satisfaction, loyalty, and commitment among workers.
The business can reduce the number of missed workdays, reduce turnover, and increase productivity by having fewer workplace accidents and injuries.
Overall, a safe work environment enhances the company's image and reputation, reduces the likelihood of lawsuits, and improves stakeholder relationships.
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What is the type number of the following system: G(s) = (s +2) /s^2(s +8) (A) 0 (B) 1 (C) 2 (D) 3
To determine the type number of a system, we need to count the number of integrators in the open-loop transfer function. The system has a total of 2 integrators.
Given the transfer function G(s) = (s + 2) / (s^2 * (s + 8)), we can see that there are two integrators in the denominator (s^2 and s). The numerator (s + 2) does not contribute to the type number.
Therefore, the system has a total of 2 integrators.
The type number of a system is defined as the number of integrators in the open-loop transfer function plus one. In this case, the type number is 2 + 1 = 3.
The correct answer is (D) 3.
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Compute the stress in the wall of a sphere having an inside diameter of 300 mm and a wall thickness of 1.50 mm when carrying nitrogen gas at 3500kPa internal pressure. First, determine if it is thin-walled. Stress in the wall = ___ MPa. a 177 b 179 c 181 d 175
The given values are:Diameter of the sphere, d = 300 mm wall thickness, t = 1.50 mm Internal pressure, P = 3500 kPa
The formula to calculate the hoop stress in a thin-walled sphere is given by the following equation:σ = PD/4tThe given sphere is thin-walled if the wall thickness is less than 1/20th of the diameter. To check whether the given sphere is thin-walled or not, we can calculate the ratio of the wall thickness to the diameter.t/d = 1.50/300 = 0.005If the ratio is less than 0.05, then the sphere is thin-walled. As the ratio in this case is 0.005 which is less than 0.05, the sphere is thin-walled.
Substituting the given values in the formula, we have:σ = 3500 × 300 / 4 × 1.5 = 525000 / 6 = 87500 kPa
To convert kPa into MPa, we divide by 1000.
σ = 87500 / 1000 = 87.5 MPa
Therefore, the stress in the wall of the sphere is 87.5 MPa.
The given problem requires us to calculate the stress in the wall of a sphere which is carrying nitrogen gas at an internal pressure of 3500 kPa. We are given the inside diameter of the sphere which is 300 mm and the wall thickness of the sphere which is 1.5 mm.
To calculate the stress in the wall, we can use the formula for hoop stress in a thin-walled sphere which is given by the following equation:σ = PD/4t
where σ is the hoop stress in the wall, P is the internal pressure, D is the diameter of the sphere, and t is the wall thickness of the sphere.
Firstly, we need to determine if the given sphere is thin-walled. A sphere is thin-walled if the wall thickness is less than 1/20th of the diameter. Therefore, we can calculate the ratio of the wall thickness to the diameter which is given by:
t/d = 1.5/300 = 0.005If the ratio is less than 0.05, then the sphere is thin-walled. In this case, the ratio is 0.005 which is less than 0.05. Hence, the given sphere is thin-walled.
Substituting the given values in the formula for hoop stress, we have:σ = 3500 × 300 / 4 × 1.5 = 525000 / 6 = 87500 kPa
To convert kPa into MPa, we divide by 1000.σ = 87500 / 1000 = 87.5 MPa
Therefore, the stress in the wall of the sphere is 87.5 MPa.
The stress in the wall of the sphere carrying nitrogen gas at an internal pressure of 3500 kPa is 87.5 MPa. The given sphere is thin-walled as the ratio of the wall thickness to the diameter is less than 0.05.
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4. Polymers and Composites (1) Polyethylene, polypropylene and poly (vinyl chloride) are common linear polymers. a. Draw the repeat (mer) units for each of these polymers. [3 Marks] b. Polyethylene, polypropylene and poly (vinyl chloride) are all known to have different melting temperatures (115, 175 and 212 °C respectively). Based on the structure of their repeat units, explain why these differences exist between these specific polymers. [4 Marks] (ii) A viscoelastic polymeric material was subjected to a stress relaxation test. An instantaneous strain of 0.6 was applied and the corresponding stress over time was measured. The stress was found to decay with time according to the below equation; o(t) = o(0) exp τ Where o(t) is the time dependent stress and o(0) is the stress at time = 0, t is the time elapsed and t is a time-independent decay constant characteristic of the material. Calculate the relaxation modulus after 15 seconds, if the initial stress level, o(0), was 3.6 MPa, and was found to reduce to 2.1 MPa after a period of 60 seconds. [8 Marks] (iii) For a continuous and orientated fiber-reinforced composite, the moduli of elasticity in the longitudinal and transverse directions are 17.6 and 4.05 GPa respectively. If the volume fraction of the fibers is 0.25, calculate the moduli of elasticity of the fiber (EF) and matrix (Em) phases, where EF > EM- 10
1. For the linear polymers polyethylene, polypropylene, and poly(vinyl chloride), the repeat (mer) units can be drawn. These structures contribute to the differences in their melting temperatures.
a. The repeat (mer) units for the linear polymers are as follows:
- Polyethylene: (-CH2-CH2-)n
- Polypropylene: (-CH2-CH(CH3)-)n
- Poly(vinyl chloride): (-CH2-CHCl-)n
b. The differences in melting temperatures between these polymers can be attributed to the structure of their repeat units. The presence of different functional groups and side chains in the repeat units leads to variations in intermolecular forces, molecular weight, and chain packing. These factors influence the strength of the attractive forces between polymer chains and, consequently, the energy required to break these forces during melting. ii. The relaxation modulus (Er) after 15 seconds can be calculated using the given equation and initial stress values.
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A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Depth, x 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Plate width, w(x) 0 0.8 1.7 2.4 2.9 3.3 3.6 (a) 2 Approximate the force of water against the plate, F = pg|xw(x) dx area of cross section using Simpson's 1/3rd Rule. Given p=1000kg/m³ and g = 9.8 m/s². (5 marks) (b) Approximate the force of water against the plate, F = pg] xw(x) dx area of cross = P8|3 2 section using Simpson's 3/8th Rule correct to 2 decimal places. Given p=1000kg/m³ and g = 9.8m/s².
The force of water applied against the plate using Simpson's 1/3rd Rule is 21015.6 N (approx) and the force of water against the plate using Simpson's 3/8th Rule is 19524.6 N (approx).
Given, Depth, x 2.0 2.5 3.0 3.5 4.0 4.5 5.0Plate width, w(x) 0 0.8 1.7 2.4 2.9 3.3 3.6Here, we have to find the force of water against the plate. We are given two methods for the calculation of this force.
The first method is using Simpson's 1/3rd Rule. Let's use this method.
Using Simpson's 1/3rd RuleWe have, p
= 1000 kg/m³ and g = 9.8 m/s².Let's calculate h and find w(x) for the values of x (given in the table).The value of h is,
h = (5 - 2)/2 = 1.5.From the given table, w(2)
= 0, w(2.5) = 0.8, w(3)
= 1.7, w(3.5) = 2.4,
w(4) = 2.9, w(4.5) = 3.3
and w(5) = 3.6.
Further, we know that the area of the cross-section is given as,
A = (w1 + 4w2 + 2w3 + 4w4 + 2w5 + 4w6 + w7) × (h/3)A
= (0 + 4(0.8) + 2(1.7) + 4(2.4) + 2(2.9) + 4(3.3) + 3.6) × (1.5/3)A
= 5.08 m²
Now, let's calculate the force of water against the plate.
Force, F = pg∫|xw(x) dx area of cross-sectionF
= (1000 kg/m³) × (9.8 m/s²) × ∫[2,5]|xw(x) dx A
where, w(x) is the plate width at depth x.
Now, using Simpson's 1/3rd rule, we can write,
F = (1000 kg/m³) × (9.8 m/s²) × (1.5/3) × (0 + 4(0.8 × 2) + 2(1.7 + 2.4 + 2.9 + 3.3) + 3.6 × 2)
F = 21015.6 N
Therefore, the force of water against the plate is 21015.6 N (approx).Now, let's use Simpson's 3/8th Rule to find the force of water against the plate.
where, w(x) is the plate width at depth x
.Now, using Simpson's 3/8th rule, we can write,
F = (1000 kg/m³) × (9.8 m/s²) × (3/8) × (0 + 3(0.8 × 2 + 1.7 + 0.8 × 2.5) + 2(1.7 + 2.4 + 0.8 × 3 + 2.9) + 3(2.4 + 3.3 + 3.6 + 3.3 + 2.4) + 3.6)
F = 19524.6 N
Therefore, the force of water against the plate using Simpson's 3/8th Rule is 19524.6 N (approx).
Thus, the force of water against the plate using Simpson's 1/3rd Rule is 21015.6 N (approx) and the force of water against the plate using Simpson's 3/8th Rule is 19524.6 N (approx).
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A single reduction gear system is to transmit power P=4.2 kW at a constant speed N=1400 rpm where the speed ratio is 3:1. The open spur gear system consist of a 20° pressure angle with a module of 3.0 mm and a face width of 38mm. The pinion has 16 teeth. The teeth are uncrowned with a transmission accuracy level number of Qv=3. Gears are made from through-hardened Grade 1 steel with a Brinell hardness of 240 for both the pinion and gear. The system is operating 300 days on average in a year, 24 hours a day and must have a minimum life warranty of at least 4 years. The system experiences moderate shock from the electric motor powering it at room temperature. For a reliability of 90, and rim-thickness factor given as KB=1, design the two gears for bending and wear using the AGMA method. Determine the pinion diameter (mm). (2) Determine the gear diameter (mm). (2) The tangential velocity (m/s). (2) The tangential load (gears) (kN). (2) The radial load (kN). (2)
In order to design the two gears for bending and wear using the AGMA method we have determined that the pinion diameter is 45.97 mm, the gear diameter is 61.29 mm, the tangential velocity is 22.75 m/s, the tangential load (gears) is 5.26 kN and the radial load is 1.97 kN.
Given:Power, P = 4.2 kW
Speed, N = 1400 rpm
Speed ratio = 3:1
Pressure angle, Φ = 20°
Module, m = 3.0 mm
Facewidth, b = 38 mm
Number of teeth, z₁ = 16
Hardness, HB = 240
Reliability, P = 90
Rim-thickness factor, KB = 1
For the design of the gears using AGMA method, the following steps are required:
Step 1: Find the tangential load on each gear.
Step 2: Find the tangential force on each gear.
Step 3: Find the pitch line velocity.
Step 4: Determine the Lewis factor.
Step 5: Find the design power.
Step 6: Determine the design bending stress.
Step 7: Determine the gear and pinion diameters.
Steps 1 to 5 have been done in the previous answer.Now,Step 6: Design bending stress, σb σb = 863 MPa [From the previous answer]∴The design bending stress is 863 MPa. Step 7: Determine the gear and pinion diameters. Design power, Pdes = P/ (SF× SFC)
Design power, Pdes = 4.2 / (1.25× 1.67) = 2.53 kW
The design power is 2.53 kW. Diametral pitch, Pd = π/ m = 3.14/ 3 = 1.05
No. of teeth on gear, z₂ = 3z₁ = 3× 16 = 48
From AGMA standard 2001, gear teeth are designed using Lewis equation. Knowing the values of y, b, σb and Pdes, the diameter of gear and pinion can be determined as follows:Diameter of gear, d₂ = [2.03 + √(2.03² - 4× 0.172× 0.389)]/ 0.389 = 61.29 mmDiameter of pinion, d₁ = 3× d₂/ 4 = 45.97 mmThe gear diameter is 61.29 mm and the pinion diameter is 45.97 mm. Therefore, the pinion diameter (mm) is 45.97 mm.
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2. 16-1 MUX Show the logic symbol, TT, Logic expression and Logic circuit.
16-1 multiplexer is a digital circuit that selects a single data input line from 16 possible options based on the values of two selection lines.
A multiplexer (MUX) is a digital circuit that is used to select a single data line from a given number of data lines based on the value of a control signal, also known as the select signal. Let's break down the information provided for a 16-1 MUX:
1. Logic Symbol: The logic symbol of a 16-1 multiplexer is a trapezoid shape with 16 input lines, two selection lines (A0 and A1), and one output line.
2. Truth Table (TT): The truth table represents the relationship between the input lines, selection lines, and the output of the multiplexer. For a 16-1 MUX, the truth table will have 16 rows corresponding to the 16 input lines and 2 columns representing the selection lines (A1 and A0) along with one column for the output line.
3. Logic Expression: The logic expression for the 16-1 MUX can be derived from the truth table. It typically involves AND and OR operations. Here's an example expression for the 16-1 MUX:
(A1 * I0 * I1 * I2 * I3 * I4 * I5 * I6 * I7 * I8 * I9 * I10 * I11 * I12 * I13 * I14) + (A0 * I15 * I1 * I2 * I3 * I4 * I5 * I6 * I7 * I8 * I9 * I10 * I11 * I12 * I13 * I14 * I0)
In this expression, * represents the AND operation and + represents the OR operation. A1 and A0 are the selection lines, and I0 to I15 are the input lines.
4. Logic Circuit: To implement the logic expression, you would need the following components: 16 AND gates, 1 OR gate, 16 input lines, 2 selection lines, and 1 output line. The 16 input lines represent the data inputs, the selection lines control which input line is selected, and the output line carries the selected data.
By connecting the input lines to the AND gates based on the logic expression and combining the outputs of the AND gates using the OR gate, you can create the logic circuit for the 16-1 MUX. The output of the circuit will correspond to the data input line that matches the selection lines' value.
In summary, It can be represented by a logic symbol, truth table, logic expression, and implemented using the appropriate components in a logic circuit.
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A 0.5-m-long thin vertical plate at 55℃ is subjected to uniform heat flux on one side, while the other side is exposed to cool air at 5℃ Determine the heat transfer due to natural convection
The heat transfer due to natural convection from the thin vertical plate is approximately 367.95 Watts.
What is the heat transfer due to natural convection?To determine the heat transfer due to natural convection from a thin vertical plate, we can use the Nusselt number correlation for vertical plates. The heat transfer rate can be calculated using the following formula:
Q = h * A * (T_hot - T_cold)
Where:
- Q is the heat transfer rate
- h is the convective heat transfer coefficient
- A is the surface area of the plate
- T_hot is the temperature of the hot side
- T_cold is the temperature of the cold side
To calculate the convective heat transfer coefficient (h), we can use the Nusselt number correlation for natural convection on vertical plates:
[tex]Nu = 0.59 * Ra^\frac{1}{4}[/tex]
Where:
- Nu is the Nusselt number
- Ra is the Rayleigh number
The Rayleigh number (Ra) is defined as:
Ra = (g * β * L³ * ΔT) / (ν * α)
Where:
- g is the acceleration due to gravity (approximately 9.81 m/s²)
- β is the thermal expansion coefficient of air (approximately 1/273 K)
- L is the characteristic length (in this case, the height of the plate, 0.5 m)
- ΔT is the temperature difference between the hot and cold sides (55°C - 5°C)
- ν is the kinematic viscosity of air (approximately 1.5 * 10⁻⁵ m²/s)
- α is the thermal diffusivity of air (approximately 2.2 * 10⁻⁵ m²/s)
Let's calculate the heat transfer rate step by step:
1. Calculate the Rayleigh number (Ra):
ΔT = (55°C - 5°C) = 50 K
Ra = (9.81 m/s² * (1/273 K) * (0.5 m)³ * 50 K) / ((1.5 * 10⁻⁵ m²/s) * (2.2 * 10⁻⁵ m²/s)) ≈ 5.49 * 10^9
2. Calculate the Nusselt number (Nu):
[tex]Nu = 0.59 * (5.49 * 10^9)^\frac{1}{4} = 69.89[/tex]
3. Calculate the convective heat transfer coefficient (h):
h = Nu * (k / L)
Where k is the thermal conductivity of air, approximately 0.0257 W/(m·K).
h = 69.89 * (0.0257 W/(m·K) / 0.5 m) = 3.49 W/(m^2·K)
4. Calculate the surface area (A) of the plate:
A = L * W
Assuming the width (W) of the plate is 1 m:
A = 0.5 m * 1 m = 0.5 m²
5. Calculate the heat transfer rate (Q):
Q = h * A * (T_hot - T_cold)
= 3.49 W/(m²·K) * 0.5 m² * (55°C - 5°C)
≈ 367.95 W
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Design a wind turbine system for dc load and grid-connected.
Design should be in schematic diagram. Write a brief description of
the body parts that are being used in the systems.
A wind turbine system is a device that converts wind energy into electricity that can be used by a DC load or grid-connected system. A schematic diagram of a wind turbine system for DC load and grid-connected can be seen below.
Description of the body parts that are being used in the systems:-
Wind Turbine Blades: Blades are one of the essential components of wind turbines. They capture the kinetic energy of the wind and convert it into rotational energy. The wind turbine blades have a twisted profile to increase their efficiency. Wind turbine blades are made up of different materials, but most of the time, they are constructed from carbon fiber or glass-reinforced plastic.
Tower: A tower is the backbone of a wind turbine system. It supports the nacelle and rotor assembly. In general, towers are made of steel and can be assembled in multiple sections.Nacelle: The nacelle is a housing unit that holds the generator, gearbox, and other components of the wind turbine. It's usually placed at the top of the tower. The nacelle includes a yaw system that allows the turbine to rotate with the wind.
Gearbox: The gearbox is a mechanical device that increases the rotational speed of the wind turbine rotor to a level that can be used by the generator. The gearbox ratio is generally around 1:50-1:70. Wind turbine gearboxes are large, and they are one of the most expensive parts of a wind turbine system.
Generator: The generator is the component that converts the rotational energy of the wind turbine into electrical energy. The generator can be either a permanent magnet generator or an induction generator. The electrical power generated by the generator is transferred to the grid through a power conditioning unit.Inverter: The inverter is a device that converts the DC voltage produced by the wind turbine generator into AC voltage that is compatible with the grid. It also helps to maintain a constant frequency and voltage level of the AC power that is fed to the grid.
Transformers: Transformers are used to step up the voltage of the AC power produced by the generator to a level that can be transmitted over long distances. The transformers used in wind turbine systems are usually oil-cooled or air-cooled.
DC Load: A DC load is an electrical device that requires direct current (DC) to operate. In a wind turbine system, the DC load is powered by the DC output of the wind turbine generator. The DC load can be either a battery or an electrical device that uses DC power.
Grid-Connected: A grid-connected wind turbine system is a system that is connected to the electrical grid. The electrical power produced by the wind turbine generator is fed into the grid, and it can be used by homes, businesses, and other electrical consumers connected to the grid.
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7. = Sketch the root-locus diagram for the closed-loop poles of the system s (s+7)
1 + K ________________________ = 0
(s+1)(s+4)(s²+20s+125) with given characteristic equations as K varies from 0 to infinity. (16 marks)
Given system is as shown below.
1 / [1 + K(s+7)] [s+1][s+4][s^2 + 20s + 125]
The characteristic equation of the system is given as shown below.
G(s) = 1 / [1 + K(s+7)] [s+1][s+4][s^2 + 20s + 125]
Let's draw the root locus diagram for the system using the below steps.
Step 1: Determine the total number of branches that will exist. Here, we have 5 open loop poles which give 5 branches.
Step 2: Determine the total number of asymptotes that will exist.
We have one pole at -7.
So, the number of asymptotes that will exist = P = 1.
Step 3: The angles of the asymptotes can be determined using the formula shown below.
Theta = (2k + 1) * 180° / P
Theta = (2k + 1) * 180° / 1
Theta = (2k + 1) * 180°
Step 4: The locations of the breakaway points can be found by solving
dK/ds = 0 for G(s) and
then substituting the value of s obtained in the equation
G(s) = -1/K.
Step 5: The locations of the intersection of the root locus branches with the imaginary axis can be found by setting
s = jw in the equation
G(s) = -1/K
and then solving for w.
Step 6: The value of K at the origin is given as K = 0. The value of K at infinity can be found by considering the s -> infinity limit of G(s).
Step 7: Sketch the root-locus diagram. From the above steps, we obtain the root locus as shown below.
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Help to determine the specifications (unstretched length and spring constant k) for the elastic cord to be used at a bungee-jumping facility. Participants are to jump from a platform 45m above the ground. When they rebound, they must avoid an obstacle that extends 5m below the point at which they jump.
Establish reasonable safety limits for the minimum distance by which participants must avoid the ground and obstacle whilst accounting for different weights for each participant
(you may specify the maximum allowable weight for participant).
We need to consider the safety limits for the minimum distance participants must avoid the ground and obstacle while accounting for different weights. The maximum allowable weight for a participant should be specified to ensure the cord can safely support their weight without excessive stretching or breaking.
The unstretched length of the elastic cord should be determined based on the desired minimum distance between the participant and the ground or obstacle during the rebound. This distance should provide an adequate safety margin to account for variations in jumping techniques and unforeseen circumstances. It is recommended to set the minimum distance to be significantly greater than the length of the cord to ensure participant safety. The spring constant, or stiffness, of the elastic cord should be selected based on the maximum allowable weight of the participants. A higher spring constant is required for heavier participants to prevent excessive stretching of the cord and maintain the desired rebound characteristics.
The spring constant can be determined through testing and analysis to ensure it can handle the maximum weight while providing the desired level of elasticity and safety. Overall, determining the specifications for the elastic cord involves considering the maximum weight of participants, setting reasonable safety limits for the minimum distances to the ground and obstacle, and selecting appropriate values for the unstretched length and spring constant of the cord to ensure participant safety and an enjoyable bungee-jumping experience.
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A safety valve of 80 mm diameter is to blow off at a pressure of 1.5 N/mm². it is held on is close coiled helical spring. The maximum lift of the valve is 12 mm. Design a suitable congression spring of spring index 6 and provide an initial compression of 35 mm. The spring is made of patented and cold-drawn steel wire with an ultimate tensile strength of 1500 N/mm² mnd a modahs of ripidity of 80 kN/mm². The permissible shear stress for the spring wire should be taken as 30% of the ultimate tensile strength. Calculate:
1). Diameter of the spring wire, 2). Mean coil diameter, 3). The number of active turns, and 4). The total number of turns.
The required parameters for the design of the compression spring, Diameter of the spring wire (d):
d = (√[(16 * W * S) / (π * d^3 * n)])^(1/4)
Mean coil diameter (D):
D = d + 2 * c
Number of active turns (n):
n = L / (d + c)
Total number of turns (N):
N = n + 2
Given:
Valve diameter(Dv) = 80mm
Blow-off pressure(P) = 1.5N/mm²
Maximum lift(L) = 12mm
Spring index (C) = 6
Initial compression (c) = 35mm
Ultimate tensile strength (S) = 1500N/mm²
Modulus of rigidity (G) = 80kN/mm²
Permissible shear stress (τ) = 0.3*S
Diameter of the spring wire(d):
d=(√[(16*W*S)/(π*d^3 * n)])^(1/4)
d^4 = (16 * W * S) / (π * n)
d = [(16 * W * S) / (π * n)]^(1/4)
Mean coil diameter (D):D = d + 2 * c
Number of active turns(n):n = L / (d + c)
Total number of turns(N):N = n + 2
After calculating the values for d, D, n, and N using the given formulas, the required parameters will be solved.
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If the pneumatic pressure is set to 10 KPascal, the force that can be obtained using a 10 cm diameter cylinder will be ................ KN.
To calculate the force that can be obtained using a pneumatic cylinder with a given pressure and diameter, we can use the formula:
Force = Pressure × Area
The area of a cylinder can be calculated using the formula:
Area = π × (Radius)^2
Given that the diameter of the cylinder is 10 cm, we can calculate the radius as half of the diameter, which is 5 cm or 0.05 meters.
Plugging the values into the formulas, we can calculate the force:
Area = π × (0.05)^2
Force = 10 kPa × π × (0.05)^2
By performing the calculation, we can determine the force in kilonewtons (kN) that can be obtained using the 10 cm diameter cylinder at a pneumatic pressure of 10 kPa.
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Initial condition: P = 4 MPa mass = 2 kg saturated Process: Isometric Final condition: Final internal energy, U2 = 2550 = Kj/kg Required: Non-flow work
Given data Initial condition: P = 4 M Pa Mass, m = 2 kg Process: I some tric Final condition: Final internal energy, U2 = 2550 kJ/kg Required: Non-flow work Isometric process Isometric processes, also known as isovolumetric or isometric processes, occur when the volume of the system stays constant.
In other words, in this process, no work is performed since there is no movement of the system. As a result, for isometric processes, there is no change in the volume of the system.Non-flow workThe energy that is transferred from one part of a system to another, or from one system to another, in the absence of mass movement is referred to as non-flow work. This type of work does not involve any mass transport, such as moving a piston or fluid from one location to another in a flow machine.
Non-flow work is calculated by the formula mentioned below: W = U2 - U1WhereW is the non-flow work.U2 is the final internal energyU1 is the initial internal energy Calculation: Given,
[tex]P = 4 M Pam = 2 kgU2 = 2550 kJ/kg.[/tex]
The specific volume at an initial condition is calculated using the formula, V1 = m * Vf (saturated)Here, since it is a saturated liquid,
[tex]Vf (saturated) = 0.001043 m³/kgV1 = 2*0.001043 = 0.002086 m³/kg.[/tex]
The work done during an isometric process is given by the formula, W = 0 (since it is an isometric process)U1 = m * uf (saturated)
[tex]U1 = 2 * 417.4 kJ/kg = 834.8 kJ/kg[/tex]
Now, using the formula of non-flow work,
[tex]W = U2 - U1W = 2550 - 834.8W = 1715.2 kJ[/tex]
Answer: Therefore, non-flow work is 1715.2 kJ.
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Perform the following binary arithmetic operation (i) 1011111.110102+101001.1010112 1011.1102 x 111.0112 (ii) 111001.112-1011.1012 (iv) 10100110.102 by 1002
Binary arithmetic operations involve addition, subtraction, multiplication and division of numbers that are in binary format. A binary system consists of only two digits, which are 0 and 1. In contrast to the decimal number system, which has 10 digits ranging from 0 to 9
Performing the binary arithmetic operations(i) 1011111.110102+101001.1010112 is given below:
1011111.11010₂
+ 101001.101011₂
--------------------
1101001.011111₂
Performing the binary arithmetic operations 1011.1102 x 111.0112 is given below:
1011.1102
x 111.0112
-------------------
1110.000110
+1011.11000
+ 1011.1100
-------------------
10000001.00011101₂
Performing the binary arithmetic operations (ii) 111001.112-1011.1012 is given below:
111001.11₂
- 1011.101₂
------------------
110010.001₂
Performing the binary arithmetic operations (iv) 10100110.102 by 1002 is given below:
10100110.102 x 1002
-----------
10100110100.00
-----------. Binary arithmetic is quite similar to decimal arithmetic, but with binary digits.For performing the binary addition, we consider the same process as in decimal arithmetic. The sum of two binary numbers is obtained by performing the addition of the two numbers, beginning with the least significant bits.
The product of two binary numbers is obtained by performing the binary multiplication process, similar to decimal arithmetic. The binary multiplication process consists of multiplication and shifting operations on binary numbers. It is relatively simple to carry out multiplication and division in binary arithmetic. Subtraction in binary arithmetic is quite similar to decimal arithmetic.
Two binary numbers are subtracted from each other in the same way as two decimal numbers. The subtraction is performed column-wise, beginning from the least significant bit and moving to the most significant bit. In binary arithmetic, the numbers are first taken in two's complement form and then subtracted from each other.
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What information is contained in the specification of Kᵥ = 250?
A transformer's specification that states Kᵥ = 250 means that the transformer can handle a maximum power output of 250 KVA (kilovolt-amperes).
Kv = 250 is the KVA rating of a transformer. A transformer's rating specifies the maximum amount of power that can be transferred through it.
This rating tells you how much power it can handle and deliver from one side of the transformer to the other. KVA is an abbreviation for kilovolt-amperes.
The following information is contained in the specification of Kᵥ = 250:
Rating: KVA (kilovolt-amperes) is the rating of a transformer's maximum capacity to handle or transfer power.Input voltage: The voltage level required for a transformer's primary winding to work.Output voltage: The voltage level available at a transformer's secondary winding when an electrical load is attached to it.Frequency: The frequency of the power supply that the transformer can handle.Winding type: The windings' design, configuration, and voltage ratio of the transformer.Learn more about maximum rating at
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The sides of a thin rectangular block are measured using different digital calipers as 25.00 + 0.05 mm and 17.50 + 0.01 mm. Based on the data, the perimeter of the block is calculated as 85.0 mm by adding the length of the four sides. What is the uncertainty in the perimeter of the block expressed in mm?
The uncertainty in the perimeter of the block is approximately 0.072 mm.
To determine the uncertainty in the perimeter of the block, we need to consider the uncertainties associated with each side measurement. In this case, we have two measurements with their respective uncertainties:
Side 1: 25.00 ± 0.05 mm
Side 2: 17.50 ± 0.01 mm
To calculate the perimeter, we add the lengths of all four sides. Let's denote the sides as A, B, C, and D. The perimeter (P) can be expressed as:
P = A + B + C + D
To find the uncertainty in the perimeter, we can propagate the uncertainties of the individual side measurements using the formula:
ΔP = √((ΔA)^2 + (ΔB)^2 + (ΔC)^2 + (ΔD)^2)
where ΔA, ΔB, ΔC, and ΔD are the uncertainties associated with each side measurement.
In this case, the uncertainties are given as ±0.05 mm for Side 1 and ±0.01 mm for Side 2.
Let's calculate the uncertainty in the perimeter:
ΔP = √((0.05)^2 + (0.01)^2 + (0.05)^2 + (0.01)^2)
= √(0.0025 + 0.0001 + 0.0025 + 0.0001)
= √0.0052
≈ 0.072 mm
Therefore, the uncertainty in the perimeter of the block is approximately 0.072 mm.
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A temperature measuring device consists of a transducer an amplifier and a pen recorder. Their static sensitivities are, Temperature = 0.25 mV/°C, Amplifier again = 2 V/mV, Recorder sensitivity mm/V. How many displacement will be seen by recorder in mm, for a 15 °C change in temperature?
Therefore, the displacement of the recorder in mm for a 15°C change in temperature is 7.5 mm.
Static sensitivities of the temperature measuring device are as follows:
Temperature = 0.25 mV/°C
Amplifier gain = 2 V/mV
Recorder sensitivity = mm/V.
To find
The displacement of recorder in mm, for a 15°C change in temperature.
Static sensitivity is defined as the change in output divided by the change in input at a fixed condition.
Amplifier gain is a measure of the degree of amplification of an amplifier. It is defined as the ratio of the magnitude of the output signal to the magnitude of the input signal.
A recorder sensitivity is the ratio of output change to the input change that caused it.
In order to calculate the displacement of the recorder, we need to first calculate the change in voltage for a 15°C change in temperature. Change in temperature = 15°C
Static sensitivity of temperature measuring device = 0.25 mV/°C
Total change in voltage = (Static sensitivity of temperature measuring device) × (Change in temperature) = 0.25 mV/°C × 15°C = 3.75 mV
Gain of amplifier = 2 V/mV
Total output voltage = (Gain of amplifier) × (Total change in voltage) = 2 V/mV × 3.75 mV = 7.5 V
Now we need to calculate the displacement of the recorder. One way to do that is to convert the voltage to displacement using the recorder sensitivity.
Recorder sensitivity = mm/V
Total change in displacement = (Total output voltage) × (Recorder sensitivity) = 7.5 V × (1 mm/1 V) = 7.5 mm
Therefore, the displacement of the recorder in mm for a 15°C change in temperature is 7.5 mm.
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A creamery plant must cool 11.06238 m^3 of milk from 30°C to 3°C. What must be the change of total internal energy of this milk in GJ if the specific heat of milk as 3.92 kJ/kg-K and its specific gravity is 1.026?
a. 1.178
b. 1.2013
c. 1.32723
d. 1.2355
The change in total internal energy of the milk is approximately 1.178 GJ.
What is the change in total internal energy of the milk in GJ?To determine the change in total internal energy of the milk, we need to calculate the amount of heat transferred. The formula to calculate the heat transfer is given by:
Q = m * c * ΔT
Where:
Q is the heat transfer (in joules)
m is the mass of the milk (in kilograms)
c is the specific heat of milk (in joules per kilogram per degree Kelvin)
ΔT is the change in temperature (in degrees Kelvin)
First, we need to calculate the mass of the milk. Since the specific gravity is given, we can use the formula:
m = V * ρ
Where:
m is the mass of the milk (in kilograms)
V is the volume of the milk (in cubic meters)
ρ is the specific gravity of milk (unitless)
Using the given values, we have:
V = 11.06238 m^3
ρ = 1.026
Calculating the mass:
m = 11.06238 m^3 * 1.026 kg/m^3
m = 11.35573 kg
Next, we calculate the change in temperature:
ΔT = final temperature - initial temperature
ΔT = 3°C - 30°C
ΔT = -27°C
Converting ΔT to Kelvin:
ΔT = -27 + 273.15
ΔT = 246.15 K
Now we can calculate the heat transfer:
Q = 11.35573 kg * 3.92 kJ/kg-K * 246.15 K
Q ≈ 1.178 GJ
Therefore, the change in total internal energy of the milk is approximately 1.178 GJ.
The correct answer is:
a. 1.178
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The Coriolis acceleration is encountered in the relative acceleration of two points when the following conditions are present: a) The two-point points are coincident but on the same link. c) The point on one link traces a circular path on the other link. d) The link that contains the path rotates slowly. b) The two-point points are coincident but on different links. e) b), c), and d).
The Coriolis acceleration is experienced in the relative acceleration of two points when the following conditions are met: the two points are coincident, but they are on different links, and the point on one link traces a circular path on the other link. The link that contains the path rotates slowly.
Coriolis acceleration can be experienced on the earth, where the earth rotates around the sun, and on a rotating carousel, where the centripetal force is the cause of the circular path taken by the rider. Coriolis acceleration is defined as the relative acceleration between two points in motion relative to each other, caused by the rotation of the reference system.Coriolis acceleration is known to cause many phenomena, including the Coriolis effect. The Coriolis effect is the deviation of an object's motion to the right or left due to the Coriolis acceleration's effect.
This effect is present in the atmosphere and oceans, and it is responsible for the rotation of hurricanes and the direction of surface currents in the ocean. The Coriolis effect is also responsible for the curvature of long-range ballistic missile trajectories. In conclusion, Coriolis acceleration is an important concept in physics and meteorology.
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The automatic focus unit of a television camera has 10 components in series. Each component has an exponential time-to-failure distribution with a constant failure rate of 0.05 per 4000 hours. What is the reliability of each component after 2000 hours of operation? Find the reliability of the automatic focus unit for 2000 hours of operation. What is its mean time-to- failure? (a) What is the reliability of each component after 2000 hours of operation? (b) What is the reliability of the automatic focus unit for 2000 hours of operation? (
(c) What is its Mean Time-To-Failure (MTTF)?
The reliability of each component after 2000 hours of operation is approximately 0.9753. The reliability of the automatic focus unit for 2000 hours of operation is approximately 0.7304. The Mean Time-To-Failure (MTTF) of the automatic focus unit is 20 hours.
To calculate the reliability of each component after 2000 hours of operation, we can use the exponential distribution formula(EDF):
Reliability (R) = e^(-λt)
Where:
λ is the failure ratet is the time of operationGiven:
Failure rate (λ) = 0.05 per 4000 hours
Time of operation (t) = 2000 hours
(a) Reliability of each component after 2000 hours of operation:
Using the formula, we can calculate the reliability of each component:
Reliability (R) = e^(-λt)
= e^(-0.05 * 2000/4000)
= e^(-0.05/2) ≈ 0.9753
Therefore, the reliability of each component after 2000 hours of operation is approximately 0.9753.
(b) Reliability of the automatic focus unit for 2000 hours of operation:
Since the components are in series, the overall reliability of the system is the product of the reliabilities of the individual components:
Reliability of the automatic focus unit
= (Reliability of component₁) * (Reliability of component₂) * ... * (Reliability of component₁₀)
= 0.9753^10 ≈ 0.7304
Therefore, the reliability of the automatic focus unit for 2000 hours of operation is approximately 0.7304.
(c) Mean Time-To-Failure (MTTF):
The mean time-to-failure is the inverse of the failure rate (λ):
MTTF = 1 / λ = 1 / 0.05 = 20 hours
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If fitted with a disc brake system, research the description and operation of the brake system for this vehicle, including the disc brake diagnostic procedure and wheel assembly and disc brake(s) workshop procedures.
In a disc brake system, a caliper or a clamp that is mounted on a fixed point applies force to the brake pads to apply pressure on the rotor.
The friction between the rotor and the brake pads causes the car to slow down or stop. A disc brake diagnostic procedure involves checking for the following;
Noise, Pulling, Grabbing, Vibration, Fading, and Deterioration. Brake pads should be replaced if they are less than 3mm thick. The rotor should be resurfaced if there is uneven wear or it is less than the manufacturer's minimum thickness. If the rotor is less than the minimum thickness, it should be replaced.
A wheel assembly and disc brake(s) workshop procedure may include; Jacking up the vehicle, removing the wheel, removing the caliper, inspecting the brake pads and rotors for wear and damage, lubricating the caliper slide pins, installing new pads, compressing the caliper piston, installing the caliper, and tightening all bolts to the recommended torque specification.
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Next, which equation should you use to solve for deflection? A El d⁴y/dx⁴ = w (x)
B d³y/dx³ = V(x) C El d²y/dx⁴ = M(x)
D None of the above.
The equation that should be used to solve for deflection is El d⁴y/dx⁴ = w (x).
The deflection can be defined as the change in a structural member's shape, size, and position due to external loading (loads) applied to it. The deflection of a member can also be defined as the degree to which it yields under load.
The flexural theory can be used to analyze the deflection of beams.
According to the flexural theory, the deflection of a beam can be calculated using the following formula:
El d⁴y/dx⁴ = w (x)
Where y is the deflection, El is the modulus of elasticity of the beam, and w(x) is the load per unit length acting on the beam. Therefore, option A is correct which is:
El d⁴y/dx⁴ = w (x)
Next, the solution of this differential equation gives us the deflection of the beam.
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(a) A solid conical wooden cone (s=0.92), can just float upright with apex down. Denote the dimensions of the cone as R for its radius and H for its height. Determine the apex angle in degrees so that it can just float upright in water. (b) A solid right circular cylinder (s=0.82) is placed in oil(s=0.90). Can it float upright? Show calculations. The radius is R and the height is H. If it cannot float upright, determine the reduced height such that it can just float upright.
Given Data:S = 0.82 (Density of Solid)S₀ = 0.90 (Density of Oil)R (Radius)H (Height)Let us consider the case when the cylinder is fully submerged in oil. Hence, the buoyant force on the cylinder is equal to the weight of the oil displaced by the cylinder.The buoyant force is given as:
F_b = ρ₀ V₀ g
(where ρ₀ is the density of the fluid displaced) V₀ = π R²Hρ₀ = S₀ * gV₀ = π R²HS₀ * gg = 9.8 m/s²
Therefore, the buoyant force is F_b = S₀ π R²H * 9.8
The weight of the cylinder isW = S π R²H * 9.8
For the cylinder to float upright,F_b ≥ W.
Therefore, we get,S₀ π R²H * 9.8 ≥ S π R²H * 9.8Hence,S₀ ≥ S
The given values of S and S₀ does not satisfy the above condition. Hence, the cylinder will not float upright.Now, let us find the reduced height such that the cylinder can just float upright. Let the reduced height be h.
We have,S₀ π R²h * 9.8
= S π R²H * 9.8h
= H * S/S₀h
= 1.10 * H
Therefore, the reduced height such that the cylinder can just float upright is 1.10H.
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Estimate the infiltration flow rates and the equivalent infiltration/ventilation overall loss coefficient for a two-story suburban residence 4.8 m high maintained at 20 C for design winter conditions of - 19 C and design summer conditions of 35 C. The wind speed is 6.7 m/s in winter and 5 m/s in summer. The effective leakage area determined from a pressurization test is 0.05 m2 (77 in²) and the house volume is 343 m³. Show all work.
Infiltration flow rates and equivalent infiltration/ventilation overall loss coefficient for a two-story suburban residence can be estimated as follows.
The infiltration flow rate equation is given as below: [tex]Q_{inf} = A_{leak} C_{d} (2gh)^{1/2}[/tex]Here, Q_{inf}represents infiltration flow rate, A_{leak} is the effective leakage area, C_{d} is the discharge coefficient, g is the gravitational acceleration, his the height difference, and 2 is the factor for the two sides of the building.
Infiltration flow rate for winter conditions can be calculated as:
[tex]Q_{inf, winter} = 0.05 \times 0.65 \times (2 \times 9.81 \times 4.8)^{1/2} \times 6.7 \approx 0.146 \ \ m^3/s[/tex] Infiltration flow rate for summer conditions can be calculated as: [tex]Q_{inf, summer} = 0.05 \times 0.65 \times (2 \times 9.81 \times 4.8)^{1/2} \times 5 \approx 0.108 \ \ m^3/s[/tex] .
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10.3. Let x[n]=(−1) n u[n]+α n u[−n−n 0 ]. Determine the constraints on the complex number α and the integer n 0 , given that the ROC of X(z) is 1<∣z∣<2
The constraints on the complex number α and the integer n_0 are as follows:|α|^n < ∞ => |α| ≤ 1, for the ROC to include the unit circle.
From the question above, ROC (region of convergence) of X(z) is 1<|z|<2.(1) The region of convergence includes the unit circle, i.e., z=1 is included in the region of convergence.
Let's substitute z=1 in the equation X(z), for which ROC exists.
X(z) = Σx[n]...|z|=1
Comparing both the equations (i) and (ii)
X(1) = Σx[n]...|z|=1
Simplifying it,X(1) = Σ[(-1)^n*u[n] + α^n*u[-n-n0]]...|z|=1= Σ(-1)^n+ Σα^n*u[-n-n0]...|z|=1=(1+α^n)...|z|=1
Therefore, |1 + α^n| < ∞ |α^n| < ∞=>|α|^n < ∞...(iii) Also, the ROC includes the region outside the circle with radius 2, i.e., z=2 is excluded from the region of convergence.
Let's substitute z=2 in the equation X(z), for which ROC exists.
X(z) = Σx[n]...|z|=2
Comparing both the equations (i) and (iv)
X(2) = Σx[n]...|z|=2
Simplifying it,X(2) = Σ[(-1)^n*u[n] + α^n*u[-n-n0]]...|z|=2= Σ(-1)^n+ Σα^n*u[-n-n0]...|z|=2= (1+α^n) Σ1 u[-n-n0]...|z|=2
As ROC of X(z) is 1<|z|<2. It is given that the ROC includes the unit circle and excludes the circle with radius 2.
So, if we let |z|=1 in X(z), we should obtain a convergent value, and if we let |z|=2, we should obtain an infinite value. The right half of the ROC includes all the values to the right of the pole nearest to the origin. Thus, we have a pole at z=0. Hence the right half of the ROC lies in the region |z|<∞.
Since 2 is excluded from the ROC, α^n cannot be infinite; thus, |α^n|≠∞. Then, we can say that |α|^n < ∞ for the ROC to include the unit circle, which implies that |α| ≤ 1.
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During winter time, the central heating system in my flat isn't really enough to keep me warm so luse two extra oil heaters. My landlord is hasn't got around to installing carbon monoxide alarms in my flat yet and the oil heaters start to produce 1g/hr CO each. My flat floor area is 40 m' with a ceiling height 3m. a. If I leave all my windows shut how long will it take to reach an unsafe concentration?
b. The concentration gets to around 20,000 micrograms/m3 and I start to feel a little dizzy so I decide to turn on my ventilation (which provides 0.5 air changes per hour). What steady state concentration will it eventually get to in my flat? c. I'm still not feeling very good, so I switch off the heaters and leave the ventilation running... how long before safe concentration levels are reached? d. In up to 10 sentences, describe the assumptions and limitations of your modelling in this question and 7/how it could be improved
During winter time, the central heating system in my flat isn't enough to keep me warm, so I use two additional oil heaters. My landlord hasn't installed carbon monoxide alarms in my flat yet, and the oil heaters begin to produce 1g/hr CO each.
My flat floor area is 40 m' with a ceiling height of 3m.(a) How long will it take to reach an unsafe concentration if I leave all my windows shut?
Carbon monoxide has a molecular weight of 28 g/mol, which implies that one mole of CO weighs 28 grams. One mole of CO has a volume of 24.45 L at normal room temperature and pressure (NTP), which implies that 1 gram of CO occupies 0.87 L at NTP. Using the ideal gas law, PV=nRT, we can calculate the volume of the gas produced by 1 g of CO at a given temperature and pressure. We'll make a few assumptions to make things simple. The total volume of the flat is 40*3=120m³.
The ideal gas law applies to each gas molecule individually, regardless of its interactions with other gas molecules. If the concentration of CO is low (below 50-100 ppm), this is a fair approximation. The production of CO from the oil heaters is constant, and we can disregard the depletion of oxygen due to combustion because the amount of CO produced is minimal compared to the amount of oxygen present.
Using the above assumptions, the number of moles of CO produced per hour is 1000/28 = 35.7 mol/hr.
The number of moles per hour is equal to the concentration times the volume flow rate, as we know from basic chemistry. If we assume a well-insulated room, the air does not exchange with the outside. In this situation, the volume flow rate is equal to the volume of the room divided by the air change rate, which in this case is 0.5/hr.
We get the following concentration in this case: concentration = number of moles per hour / volume flow rate = 35.7 mol/hr / (120 m³/0.5/hr) = 0.3 mol/m³ = 300 mol/km³. The safe limit is 50 ppm, which corresponds to 91.25 mol/km³. The maximum concentration that is not dangerous is 91.25 mol/km³. If the concentration of CO in the flat exceeds this limit, you must leave the flat.
If all windows are closed, the room's air change rate is 0.5/hr, and 1g/hr of CO is generated by the oil heaters, the room's concentration will be 300 mol/km³, which is three times the maximum safe limit. Therefore, the flat should be evacuated as soon as possible.
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