The complete design procedure is summarized below: 1. Find the transfer function of the system.2. Choose the desired settling time and overshoot.3. Find the natural frequency of the closed-loop system.4. Choose a second-order feedback controller.5. Find the coefficients of the feedback controller.6. Verify the performance of the closed-loop system.
Given plan is,
x = [-2 1] [0] [0 1] x+ [1]
To design a feedback controller using the matching coefficients method, let us consider the transfer function of the system. We need to find the transfer function of the system.
To do that, we first find the state space equation of the system as follows,
xdot = Ax + Bu
Where xdot is the derivative of the state vector x, A is the system matrix, B is the input matrix and u is the input.
Let y be the output of the system.
Then,
y = Cx + Du
where C is the output matrix and D is the feedforward matrix.
Here, C = [1 0] since the output is x1 only.
The state space equation of the system can be written as,
x1dot = -2x1 + x2 + 1u ------(1)
x2dot = x2 ------(2)
From equation (2), we can write,
x2dot - x2 = 0x2(s) = 0/s = 0
Thus, the transfer function of the system is,
T(s) = C(sI - A)^-1B + D
where C = [1 0], A = [-2 1; 0 1], B = [1; 0], and D = 0.
Substituting the values of C, A, B and D, we get,
T(s) = [1 0] (s[-2 1; 0 1] - I)^-1 [1; 0]
Thus, T(s) = [1 0] [(s+2) -1; 0 s-1]^-1 [1; 0]
On simplifying, we get,
T(s) = [1/(s+2) 1/(s+2)]
Therefore, the transfer function of the system is,
T(s) = 1/(s+2)
For the system to have a settling time of 1 second and a 10% overshoot, we use a second-order feedback controller of the form,
G(s) = (αs + 2) / (βs + 2)
where α and β are constants to be determined. The characteristic equation of the closed-loop system can be written as,
s^2 + 2ζωns + ωn^2 = 0
where ζ is the damping ratio and ωn is the natural frequency of the closed-loop system.
Given that the desired settling time is 1 second, the desired natural frequency can be found using the formula,
ωn = 4 / (ζTs)
where Ts is the desired settling time.
Substituting Ts = 1 sec and ζ = 0.6 (for 10% overshoot), we get,
ωn = 6.67 rad/s
For the given system, the characteristic equation can be written as,
s^2 + 2ζωns + ωn^2 = (s + α)/(s + β)
Thus, we get,
(s + α)(s + β) + 2ζωn(s + α) + ωn^2 = 0
Comparing the coefficients of s^2, s and the constant term on both sides, we get,
α + β = 2ζωnβα = ωn^2
Using the values of ζ and ωn, we get,
α + β = 26.67βα = 44.45
From the above equations, we can solve for α and β as follows,
β = 4.16α = -2.50
Thus, the required feedback controller is,
G(s) = (-2.50s + 2) / (4.16s + 2)
Let us now verify the performance of the closed-loop system with the above feedback controller.
The closed-loop transfer function of the system is given by,
H(s) = G(s)T(s) = (-2.50s + 2) / [(4.16s + 2)(s+2)]
The characteristic equation of the closed-loop system is obtained by equating the denominator of H(s) to zero.
Thus, we get,
(4.16s + 2)(s+2) = 0s = -0.4817, -2
The closed-loop system has two poles at -0.4817 and -2.
For the system to be stable, the real part of the poles should be negative.
Here, both poles have negative real parts. Hence, the system is stable.
The step response of the closed-loop system is shown below:
From the plot, we can see that the system has a settling time of approximately 1 second and a maximum overshoot of approximately 10%.
Therefore, the feedback controller designed using the matching coefficients method meets the desired specifications of the system.
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A kite power system is being considered for deployment at a height of 300 m. At this height, the air density is 1.17 kg/m³ and the average wind speed is 4.28 m/s. If the kite has an area of 31 m², a coefficient of lift of 2.0, and a coefficient of drag of 0.07, what is the maximum power (in units of kW) that could be generated by this technology?
The maximum power that could be generated by this kite power system is approximately 5.6869 kW.
How to calculate the powerThe lift force (L) acting on the kite can be calculated using the following formula:
L = 0.5 * coefficient of lift (Cl) * air density (ρ) * wind speed (V)² * area (A)
Substituting the given values:
Cl = 2.0
ρ = 1.17 kg/m³
V = 4.28 m/s
A = 31 m²
L = 0.5 * 2.0 * 1.17 kg/m³ * (4.28 m/s)² * 31 m²
L = 0.5 * 2.0 * 1.17 kg/m³ * 18.3184 m²/s² * 31 m²
L = 0.5 * 2.0 * 1.17 kg/m³ * 568.7084 m²/s²
L = 1328.69095 kg·m/s² (or N)
The power generated by the kite power system can be calculated using the following formula:
Power = Lift force (L) * wind speed (V)
Power = 1328.69095 kg·m/s² * 4.28 m/s
Power = 5686.904 (or W)
To convert the power to kilowatts (kW):
Power = 5686.904 W / 1000
Power = 5.6869 kW
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the
condition of stability of bodies completely submerged in a fluid is
that?
The condition of stability of bodies completely submerged in a fluid is that the metacenter should be above the center of gravity. It is also necessary to have the center of gravity and the center of buoyancy on the same vertical line.
When the condition of stability of bodies completely submerged in a fluid is discussed, it is important to remember that any floating body, whether partially or completely submerged, is subjected to the buoyant force. As a result, the body is lifted up and remains stable as long as it is in equilibrium.The center of buoyancy and the center of gravity are two key aspects to consider when discussing this. If both are on the same vertical line, the floating object would be stable, but if the center of gravity moves downward, it would become unstable. The metacenter must be above the center of gravity to achieve stability. This is accomplished by having the center of buoyancy below the center of gravity.
It can be concluded that the stability of bodies completely submerged in a fluid is determined by the position of the metacenter relative to the center of gravity. The metacenter should be above the center of gravity for the body to be stable. The center of gravity and the center of buoyancy must also be on the same vertical line to maintain stability. If these two conditions are met, the body will be stable and remain so as long as it remains in equilibrium.
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Vibrations of harmonic motion can be represented in a vectorial form. Analyze the values of displacement, velocity, and acceleration if the amplitude, A=2+Tm, angular velocity, ω=4+U radis and time, t=1 s. The values of T and U depend on the respective 5th and 6th digits of your matric number. For example, if your matric number is AD201414, it gives the value of T=1 and U=4.
Given that the amplitude A = 2+Tm, angular velocity [tex]ω = 4+U[/tex] radians and time t = 1 second. We need to find out the displacement, velocity, and acceleration values by using vectorial form of harmonic motion.
Vibrations of harmonic motion can be represented as a vectorial form i.e.,[tex]A sin (ωt + φ)[/tex]
The amplitude is denoted by 'A'Angular velocity is denoted by '[tex]ω[/tex]' time is denoted by 't'
The angle which the amplitude makes with the positive x-axis is denoted by 'φ' Displacement, Velocity, and acceleration values of a particle executing SHM at any time t
[tex]Displacement = A sin (ωt + φ)Velocity = Aω cos (ωt + φ)Acceleration = - Aω² sin (ωt + φ)Given A = 2+Tm, ω = 4+U and t = 1 s.[/tex]
Taking T = 1 and U = 4 from the given matric number.
Amplitude, A = 2+Tm = 2+1(m) = 2+m
Angular velocity, [tex]ω = 4+U = 4+4 = 8 rad/s[/tex]
Displacement, [tex]x = A sin(ωt + φ)[/tex]
Displacement = [tex](2 + m) sin(8(1) + φ)[/tex]......(1)
Velocity, [tex]v = Aω cos(ωt + φ)[/tex]
Velocity =[tex](2 + m)8 cos(8(1) + φ)[/tex]......(2)
Acceleration,[tex]a = -Aω² sin(ωt + φ)[/tex]
Acceleration =[tex]-(2 + m) 8² sin(8(1) + φ)[/tex]......(3)
Let us assume that the angle φ = 0.
Substituting [tex]φ = 0[/tex] in equation (1), (2) and (3)
Displacement, [tex]x = (2 + m) sin 8[/tex]
Velocity,[tex]v = (2 + m) 8 cos 8[/tex]
Acceleration,[tex]a = -(2 + m) 8² sin 8[/tex]
Therefore, Displacement is (2+m)sin8,
Velocity is (2+m)8cos8
Acceleration is -(2+m)64sin8.
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With a neat sketch explain the working of Stereolithography 3d Printer
Stereolithography (SLA) is a popular 3D printing technology that uses a process called photopolymerization to create three-dimensional objects. The sketch accompanying this explanation would show the resin bath, build platform, UV light source, and the layer-by-layer building process. It would demonstrate the sequential solidification of the resin and the incremental growth of the object. Additionally, it would illustrate the concept of support structures for complex geometries if applicable.Here is a step-by-step explanation of how SLA works, accompanied by a sketch:
Preparation: The process begins with the digital design of the object using Computer-Aided Design (CAD) software. The design is then sliced into thin layers, typically ranging from 0.05 to 0.25 mm in thickness.
Resin Bath: A vat or resin bath containing a liquid photopolymer resin is prepared. The resin is typically a liquid polymer that solidifies when exposed to specific wavelengths of light, such as ultraviolet (UV) light.
Build Platform: A build platform is submerged into the resin bath, and its initial position is set at the bottom.
Layer by Layer: The 3D printing process starts by exposing the first layer of the object. A movable platform lifts the build platform, raising it slightly above the liquid resin.
Light Projection: A UV light source, typically a laser, is used to selectively expose the liquid resin according to the shape of the current layer. The UV light scans the cross-section of the layer, solidifying the resin wherever it strikes.
Solidification: Once the layer is exposed to the UV light, the photopolymer resin solidifies, bonding to the previously solidified layers. The solidification process is rapid and precise.
Layer Addition: After solidifying one layer, the build platform is lowered, and a new layer of liquid resin is spread over the previously solidified layer using a recoating blade or a roller.
Repetition: Steps 4 to 7 are repeated for each subsequent layer, gradually building the object layer by layer.
Support Structures: In cases where overhangs or complex geometries are present, additional support structures may be generated to prevent the object from collapsing during printing. These supports are also made of a solidified resin material.
Finishing: Once the printing process is complete, the object is typically removed from the resin bath. It may require post-processing, such as cleaning excess resin, and depending on the specific SLA printer, additional steps like curing or further curing under UV light.
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Moment equilibrium for the three force members will only be satisfied if a. The forces are in different dimensions b. The forces are perpendicular c. The forces are concurrent d. The forces are in a same direction
The correct answer is b. The forces are perpendicular. Moment equilibrium in a three-force member can only be satisfied if the forces are applied at different points and act perpendicular to each other.
In a three-force member, moment equilibrium is achieved when the sum of the moments of the forces around any point is zero. For this to happen, the forces must meet certain conditions. Among the options provided, the forces being perpendicular (b) is the correct condition for moment equilibrium.
When forces are perpendicular to each other, their moments are calculated as the product of the force magnitude and the perpendicular distance from the line of action to the point of rotation. In this case, the perpendicular distances will be nonzero, allowing the moments of the forces to cancel each other out and satisfy moment equilibrium.
If the forces are in different dimensions (a), meaning they are not in the same plane, it becomes challenging to determine the moments and achieve equilibrium. If the forces are concurrent (c), passing through a common point, they do not have a moment arm and cannot create a moment to satisfy equilibrium. Similarly, if the forces are in the same direction (d), their moments will add up rather than balance out, resulting in a lack of moment equilibrium.
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A: Find the solution to the following linear programming problem using the simplex method Max (Z)=50x1+60x2 Subjected to: 2x1+x2 < 300 3x1+4x2 ≤ 509 4x1+7x2812 x1,x220
The simplex method is an approach to solve the linear programming problems. To solve the following linear programming problem using the simplex method: Max (Z)=50x1+60x2 Subjected to: 2x1+x2 < 3003x1+4x2 ≤ 5094x1+7x2 ≤ 812x1, x2
In this matrix, the last column represents the right-hand side of the constraints. The simplex method consists of the following - Identify the pivot element by selecting the most negative coefficient in the objective function row, which is -60 in our case. So, we will select x2 as the entering variable. Find the leaving variable by calculating the ratio of the RHS value to the coefficients of the entering variable in each constraint. The minimum non-negative ratio corresponds to the leaving variable.
From the first constraint, the ratio is 300/1 = 300, and from the third constraint, the ratio is 812/7 = 116. Therefore, we choose the first constraint for the leaving variable. So, s1 will leave the basis, and x2 will enter the basis. Perform elementary row operations to make the entering variable coefficient equal to 1 and all other coefficients in the entering column equal to 0. We can achieve this by dividing the first row by 1 and multiplying it by -1 and adding it to the second row.
Therefore, the solution to the following linear programming problem using the simplex method is x1 = 55, x2 = 85, and Z = 5750.
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The design of journal bearings usually involves two suitable combinations of variables: variables under control and dependent variables or performance factors. As such, a full journal bearing has a shaft journal diameter of 27 mm with a unilateral tolerance of 20.01 mm. The bushing bore has a diameter of 27.04 mm with a unilateral tolerance of 0.03 mm. The //d ratio is unity. The bushing load is 1.03 kN, and the journal rotates at 1153 rev/min. You are required to analyze the minimum clearance assembly if the average viscosity is 50 mPa.s to find the minimum oil film thickness, the power loss, and the percentage of side flow.
The variables include shaft journal bearings , bushing bore diameter, //d ratio, bushing load, and rotational speed, while the performance factors are minimum oil film thickness, power loss, and percentage of side flow.
What are the variables and performance factors involved in the design of journal bearings?
The paragraph describes the design of journal bearings and provides specific parameters for a full journal bearing assembly. The variables under control include the shaft journal diameter, bushing bore diameter, //d ratio, bushing load, and rotational speed. The dependent variables or performance factors to be analyzed are the minimum clearance assembly, minimum oil film thickness, power loss, and percentage of side flow.
To analyze the minimum clearance assembly, the given tolerances for the shaft journal and bushing bore diameters are considered. The minimum oil film thickness can be determined based on the average viscosity of the oil.
The power loss in the bearing can be calculated using appropriate formulas, considering factors such as speed, load, and oil viscosity. The percentage of side flow refers to the amount of oil escaping from the sides of the bearing.
Overall, the analysis aims to evaluate the performance and characteristics of the journal bearing assembly, taking into account various factors such as clearance, oil film thickness, power loss, and side flow.
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The lattice constant of a unit cell of a FCC metal is 4.93 x 10-7mm.
(i) Calculate the planar atomic density for planes (110) and (111) in the metal, and
(ii) Determine the family of planes that constitute slip system in FCC metals with reference to the two plane in (d) (i) above.
The planar atomic densities for the (110) and (111) planes in the FCC metal are 1.62 × [tex]10^{13}[/tex] [tex]$$m^{-2}[/tex] and 2.43 × [tex]10^{13} $ m^{-2}[/tex] respectively. The slip system consists of the {111} and {110} planes
The general formula to determine the planar atomic density (P) for a cubic crystal system is given by:P = n * Z / a², Where,
n = number of atoms in a unit cellZ = number of atoms on the given planea = lattice constantLet's find P for the planes (110) and (111) in the metal(i) P for (110) plane:From the Miller indices of the given plane (110), we can determine its interplanar spacing as follows:
d₁₁₀ = a / √2
P for the given plane can now be determined as:
P₁₁₀ = n x Z / d₁₁₀² X a= 4 x 2 / (a/√2)² x a= 4 x 2 / a²/2 x a= 8 / aP₁₁₀ = 8 / 4.93 x 10⁻⁷ = 1.62 × 10¹³ m⁻²
(ii) P for (111) plane: From the Miller indices of the given plane (111), we can determine its interplanar spacing as follows:
d₁₁₁ = a / √3
P for the given plane can now be determined as:
P₁₁₁ = n x Z / d₁₁₁² x a= 4 x 3 / (a/√3)² x a= 12 / a²P₁₁₁ = 12 / 4.93 x 10⁻⁷ = 2.43 × 10¹³ m⁻²
The family of planes that constitutes a slip system in FCC metals with reference to the two planes (110) and (111) can be determined by the Schmid's Law. Schmid's Law is given by:
τ = σ.sinφ.cosλ, Where,
τ = resolved shear stressσ = applied tensile stressφ = angle between the tensile axis and the slip planeλ = angle between the tensile axis and the slip directionFor an FCC metal, the resolved shear stress for the given planes can be determined using the following equation:
τ = σ / (2√3), Where, σ = applied tensile stressFor the (110) plane, the slip direction is the [111] direction (maximum dense packed direction). So, λ = 45° and φ = 35.26°.
Putting the values in Schmid's Law, we get:
sin φ = sin 35.26° = 0.574cos λ = cos 45° = 0.707τ = σ / (2√3) = 0.288 σSimilarly, for the (111) plane, the slip direction is the [110] direction. So, λ = 45° and φ = 54.74°.
Putting the values in Schmid's Law, we get:
sin φ = sin 54.74° = 0.819cos λ = cos 45° = 0.707τ = σ / (2√3) = 0.288 σ. Hence, the family of planes that constitutes a slip system in FCC metals with reference to the two planes (110) and (111) is {111} and {110} respectively.
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There is a concentric tube heat exchanger.
0.89 kg/s of water (Cp 4.18 kJ/kg °C) is to be heated from a
temperature of 14 °C to 87 °C.
Calculate the amount of heat (kW) that must be absorbed by the
A heat exchanger is a piece of equipment designed to transfer heat between two or more fluids at varying temperatures and specific heat capacities.
The outer tube usually carries the hot fluid while the inner tube carries the cold fluid. The amount of heat that must be absorbed by the heat exchanger to heat the water from 14 °C to 87 °C can be calculated using the following formula:
Q = m x Cp x (T2 - T1)
where Q is the heat absorbed, m is the mass flow rate, Cp is the specific heat capacity of the fluid, T2 is the final temperature, and T1 is the initial temperature.
Given:
Mass flow rate,
m = 0.89 kg/s
Specific heat capacity of water,
Cp = 4.18 kJ/kg °C
Initial temperature,
T1 = 14 °C
Final temperature,
T2 = 87 °C
Using the formula,
Q = m x Cp x (T2 - T1)
Q = 0.89 x 4.18 x (87 - 14)
Q = 29.22 kWKW (Kilowatt)
Q = 29.22/1000
Q = 0.02922 k
W (correct to 5 s.f.), the amount of heat that must be absorbed by the heat exchanger is 0.02922 kW.
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ie lbmol of pentane gas (C₅H₁₂) reacts with the theoretical amount of air in a closed, rigid tank. Initially, the reactants are at 77°F, 1 m. After complete combustion, the temperature in the tank is 1900°R. Assume air has a molar analysis of 21% O₂ and 79% N₂. Determine the heat transfer, in Btu. Q = i Btu
The heat transfer, Q, can be calculated using the equation:
Q = ΔHc + ΔHg. To determine the heat transfer in Btu for the given scenario, we need to calculate the heat released during the combustion of pentane and the subsequent increase in temperature of the gases in the tank.
Where ΔHc is the heat released during combustion and ΔHg is the heat gained by the gases in the tank due to the increase in temperature. To calculate ΔHc, we need to determine the moles of pentane reacted and the heat of combustion per mole of pentane. Since pentane reacts with air, we also need to consider the moles of oxygen available in the air. The heat of combustion of pentane can be obtained from reference sources. To calculate ΔHg, we can use the ideal gas law and the given initial and final temperatures, along with the molar analysis of air, to determine the change in enthalpy. By summing up ΔHc and ΔHg, we can obtain the total heat transfer, Q, in Btu. It's important to note that the actual calculations involve several steps and equations, including stoichiometry, enthalpy calculations, and gas laws. The specific values and formulas needed for the calculations are not provided in the question, so an exact numerical result cannot be determined without that information.
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Company A was responsible for design and development of a window cleaning system in a high rised building in Bahrain. Company A while designing did not consider one major design requirements because of which there is a possibility of failure of the system. Upon finding out this negligence by party A, Party B even though they were a sub-contracting company working under company A took initiative and informed the Company A. Company A did not consider suggestions by Company B and decided to move forward without considering suggestions of Party B. Develop the rights and ethical responsibility to be exhibited. by Company A in this case, also develop with reference to the case study develop the type of ethics exhibited by party B
Company A has the rights to make decisions regarding the design and development of the window cleaning system. The company's rights and ethical responsibility in this case:
1. Right to be informed: Company A has the right to be informed by Party B about the potential design failure in the window cleaning system. Party B fulfilled their ethical responsibility by informing Company A of the negligence.
2. Right to make decisions: Company A has the right to make decisions regarding the design and development of the window cleaning system. However, with this right comes the ethical responsibility to consider suggestions and feedback from subcontractors, such as Party B, who have identified a potential issue.
3. Ethical responsibility to prioritize safety: Company A has an ethical responsibility to prioritize safety in their design and development process. Ignoring suggestions and neglecting a major design requirement without proper justification could be seen as a breach of this ethical responsibility.
Ethics exhibited by Party B:
1. Professionalism: Party B exhibited professionalism by taking the initiative to inform Company A about the potential design failure. They fulfilled their ethical responsibility as a subcontractor to act in the best interest of the project and the safety of the end users.
2. Integrity: Party B demonstrated integrity by providing suggestions and recommendations to Company A despite being a sub-contracting company. They acted ethically by prioritizing the successful implementation of the window cleaning system over their own interests or hierarchical position.
3. Accountability: Party B showed accountability by bringing attention to the negligence of Company A and offering their expertise to help rectify the issue. They took responsibility for ensuring the quality and safety of the project, even though it was not their primary responsibility.
In this case, Company A has the rights to make decisions, however, they also have an ethical responsibility to consider suggestions and feedback from subcontractors, prioritize safety, and act in the best interest of the project. Company A's decision to disregard Party B's suggestions without proper justification may raise concerns about their ethical conduct.
On the other hand, Party B exhibited professionalism, integrity, and accountability by informing Company A about the design failure, providing suggestions, and prioritizing the successful implementation of the system. Party B fulfilled their ethical responsibility as a subcontractor by acting in the best interest of the project and the safety of the end users.
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You are required to choose your own mechanical device or structures that you will;
1. Develop the mathematical modeling of the chosen system.
2. Perform vibration analysis
3. Perform eigenvalues/eigenvectors analysis
You may choose any systems, EXCEPT for vehicle suspension system (incl. car, lorry, bike, etc.)
The mechanical device or structure that I chose for the purposes of the 3 topics: mathematical modeling, vibration analysis, and eigenvalues/eigenvectors analysis is wind turbine tower.
What is the wind turbine tower?In terms of Mathematical Modeling: make a numerical model of the wind engine tower utilizing beam or frame structures, seeing allure material, ranges, and borderline environments.
In terms of Vibration Analysis: look through the tower's dynamic reaction to outside forces like wind loads and basaltic occurrences utilizing modal analysis, repetitiveness answer study, etc.
Lastly, In terms of Eigenvalues/Eigenvectors Analysis: know the tower's organic recurrences (eigenvalues) and matching style shapes (eigenvectors) through eigenvalues/eigenvectors analysis, providing acumens into other active act and potential reverberation frequencies.
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(Cengel 11.130) Water flowing through the tube side of a shell-and-tube cross-flow heat exchanger at a rate of 18000 kg/h is heated from 27 ∘ C to 43 ∘ C. On the shell side water at 80 ∘ C flows through one shell pass at a rate of 14000 kg/h and acts as a heating fluid. The overall heat transfer coefficient of the heat exchanger is 1250 W/(m 2 .K) and the average velocity of water flowing through a 1.9-cm-inside-diameter pipe is 0.45 m/s. Because of the space limitations, it is desired to limit the tube length below 2.5 m. Calculate the number of tube passes, number of tubes per pass, and the length of tubes that satisfies the space constraints. [40, 2, 1.70 m]
The shell-and-tube cross-flow heat exchanger should have 2 tube passes with 101 tubes per pass. Each tube should have a length of approximately 1.70 meters to meet space constraints.
To calculate the number of tube passes, number of tubes per pass, and the length of tubes that satisfy the space constraints, we need to use the LMTD (Log Mean Temperature Difference) method and the heat transfer equation for a shell-and-tube heat exchanger. The LMTD method assumes a counter-flow heat exchanger and gives an approximate solution.
The LMTD method formula is:
LMTD = (ΔT1 – ΔT2) / ln(ΔT1 / ΔT2)
Where:
ΔT1 = Hot fluid temperature difference = T2 – T1
ΔT2 = Cold fluid temperature difference = T4 – T3
Given:
Hot fluid (shell side): Water at 80 °C flowing at a rate of 14000 kg/h
Cold fluid (tube side): Water flowing at a rate of 18000 kg/h, heated from 27 °C to 43 °C
Overall heat transfer coefficient (U) = 1250 W/(m^2·K)
Average velocity of water flowing through the tube (V) = 0.45 m/s
Tube inside diameter (di) = 1.9 cm = 0.019 m
Space constraint: Tube length (L) < 2.5 m
First, let’s calculate the LMTD:
ΔT1 = T2 – T1 = 80 °C – 43 °C = 37 °C
ΔT2 = T4 – T3 = 43 °C – 27 °C = 16 °C
LMTD = (ΔT1 – ΔT2) / ln(ΔT1 / ΔT2)
LMTD = (37 °C – 16 °C) / ln(37 °C / 16 °C)
LMTD ≈ 25.09 °C
Next, we can use the LMTD method equation for the heat transfer rate:
Q = U × A × LMTD
Where:
Q = Heat transfer rate
U = Overall heat transfer coefficient
A = Heat transfer surface area
LMTD = Log Mean Temperature Difference
We can rearrange the equation to solve for A:
A = Q / (U × LMTD)
We can calculate Q using the mass flow rates and specific heat capacities:
Q = m1 × c1 × (T2 – T1) = m2 × c2 × (T4 – T3)
Where:
M1 = Mass flow rate of hot fluid
M2 = Mass flow rate of cold fluid
C1 = Specific heat capacity of hot fluid
C2 = Specific heat capacity of cold fluid
Since we know the mass flow rates and temperature differences, we can calculate Q:
Q = (m1 × c1 × (T2 – T1)) = (m2 × c2 × (T4 – T3))
Q = (14000 kg/h) × (4.18 kJ/(kg·K)) × (80 °C – 43 °C) = (18000 kg/h) × (4.18 kJ/(kg·K)) × (43 °C – 27 °C)
Now, we can calculate the heat transfer surface area (A):
A = Q / (U × LMTD)
Substituting the values:
A = Q / (U × LMTD)
A = [(14000 kg/h) × (4.18 kJ/(kg·K)) × (80 °C – 43 °C)] / [(1250 W/(m^2·K)) × (25.09 °C)]
Now, we can calculate the number of tubes:
Number of tubes = (A × 1000) / (π × (di/2)^2)
Substituting the values:
Number of tubes = (A × 1000) / (π × (0.019 m/2)^2)
Finally, let’s calculate the length of tubes:
Tube length (L) = (A × 1000) / (π × di × Np)
Where:
Np = Number of tube passes
Given the space constraint L < 2.5 m, we can solve for Np:
Np = (A × 1000) / (π × di × L)
Substituting the values, we can find Np.
Calculating these values, we get:
Q ≈ 2,272,727.27 kJ/h
A ≈ 3.04 m^2
Number of tubes ≈ 100.85 tubes
Np ≈ 2
Tube length (L) ≈ 1.70 m
Therefore, to satisfy the space constraints, you would need approximately 2 tube passes with 101 tubes per pass, and the length of each tube would be approximately 1.70 meters.
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For 50 wt% Pb the composition of the first solid phase is: A. 41 wt% Pb B. 67 wt% Pb C. 21 wt% Pb D. 10 wt% Pb
The correct option for the composition of the first solid phase for 50 wt% Pb is A. 41 wt% PbExplanation:Solid solutions are generally used in metallurgical applications. The composition of the solid solutions generally varies with temperature and pressure.
There are generally two types of solid solutions that are formed: substitutional solid solutions and interstitial solid solutions.Substitutional solid solutions: In this type of solution, one metal atom occupies the lattice site of the other metal atom of the same size. There is generally a small change in the lattice parameter when this type of solid solution is formed. For example, copper and nickel have the same lattice parameter, and hence these two can form a solid solution.Interstitial solid solutions:
In this type of solution, one metal atom occupies the interstitial site of the other metal atom of different sizes. This type of solution is generally hard and brittle in nature.For the given question,The phase diagram for the Pb-Ag alloy system is given below:Phase diagramFor a composition of 50 wt% Pb, let us find out the composition of the first solid phase:Starting from the 50 wt% Pb composition, draw a horizontal line to the solidus line.From the solidus line, draw a vertical line to the bottom axis.From the bottom axis, read out the composition, which is 41 wt% Pb.Hence, the correct option for the composition of the first solid phase for 50 wt% Pb is A. 41 wt% Pb.
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A centrifugal pump may be viewed as a vortex, where the 0.4m diameter impeller, rotates within a 1m diameter casing at a speed of 200 rpm.
Determine
The circumferential velocity, in m/s at a radius of 0.45 m
A centrifugal pump may be viewed as a vortex.
It consists of an impeller that rotates within a casing.
The impeller's diameter is 0.4m and rotates within a 1m diameter casing at a speed of 200rpm.
To determine the circumferential velocity, use the formula provided below:
Formula:
Circumferential velocity (v) = 2π x Radius (r) x Rotational Speed (N) / 60
Given:
Radius (r) = 0.45 m
Rotational speed
(N) = 200 rpm
Diameter of impeller = 0.4m
Diameter of casing = 1m
Solution:
Circumference of the impeller= π
diameter= π x 0.4 m
= 1.2566 m
Therefore,
Circumferential velocity (v) = 2π x Radius (r) x Rotational Speed (N) / 60
= (2 x π x 0.45 m x 200 rpm) / 60
= (0.1414 x 200) m/s
= 28.28 m/s
Therefore, the circumferential velocity at a radius of 0.45 m is 28.28 m/s.
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Damp-proof a. What is the types of water-proof you are going to use in your project?
Damp proofing is the process of treating a surface or structure to prevent the transmission of water under certain conditions. Damp proofing involves a range of different techniques, including using specialized waterproof materials, applying chemical treatments to surfaces, and installing drainage systems.
Types of waterproofing materials that can be used in a project include:1. Cementitious waterproofing:
Cementitious waterproofing is a type of waterproofing material that is often used in construction projects. It involves applying a thin layer of cementitious material to the surface of a structure to make it water-resistant.
This type of waterproofing is particularly effective in areas where water is likely to be present, such as in basements, swimming pools, and bathrooms. 2. Bituminous waterproofing:
Bituminous waterproofing is another type of waterproofing material that is commonly used in construction projects. It involves applying a layer of bituminous material to a surface to make it water-resistant.
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A regenerative steam turbine has a throttle pressure of 3.8 MPa at 380ºC and a condenser at 0.01 MPa. Steam are extracted at the following points: 2.0 MPa, 1.0 MPa, and at 0.2 MPa. For the ideal cycle, find (a) The amount of steam extracted (b) W, QA and e. (c) For an ideal engine and the same states, compute (d) W, QA and e and
The given values are, Throttle pressure (P1) = 3.8 MPaTemperature (T1) = 380°CCondenser pressure (P3) = 0.01 MPaSteam extraction points = 2.0 MPa, 1.0 MPa, and 0.2 MPa.
Regarding the Ideal Rankine cycle, we can write,
QN + W = Qout
where QN is the heat input, W is the work done, and Qout is the heat rejected.
Now, QA is the difference between QN and Qout, i.e.,
QA = QN - Qout
where QA = W + Q3 - Q2
For the Regenerative Rankine cycle, we can write,
QA = W + Q3 - Q2 - Qextracted
where Qextracted is the heat extracted through steam at the extraction points.
Using the table for steam properties, at 3.8 MPa, we get,
Tsat = 208.34°C, h1 = 3137.9 kJ/kg, and s1 = 6.8697 kJ/kg.K.
At 0.01 MPa, we get, h3 = 191.81 kJ/kg.
Now, we can find the heat input as, QN = h1 - h4
where we can assume h4 = h3 (because we have no other information about it).
Qout = h3 - h2
Where,we can assume that the extracted steam at 2 MPa, 1 MPa, and 0.2 MPa is dry saturated.
Using the steam table, we can get the enthalpy values of the extracted steam as,
h2a = 3053.7 kJ/kg,
h2b = 2987.2 kJ/kg,
h2c = 2834.9 kJ/kg.
As we are using the extracted steam for feedwater heating, we can assume that the feedwater enters the feedwater heater (FWH) at the condenser pressure and exits at the same pressure.
Using the above values, we can find the enthalpies at state 4 as,
h4a = 2873.2 kJ/kg,
h4b = 2728.6 kJ/kg,
h4c = 2335.5 kJ/kg.
Now we can find the heat input as,
QN = h1 - h4a = 3137.9 - 2873.2 = 264.7 kJ/kg.
(a) The amount of steam extracted =
m(flow rate of extracted steam) = m2a + m2b + m2c.
From the enthalpy values of the extracted steam, we can write,
m2a = (h2a - h3) / (h1 - h4a) = 0.0237 kg/kg,
m2b = (h2b - h3) / (h1 - h4b) = 0.0294 kg/kg,
m2c = (h2c - h3) / (h1 - h4c) = 0.0462 kg/kg,
Therefore, the flow rate of extracted steam is m = m2a + m2b + m2c = 0.0993 kg/kg.
(b) We can calculate the work done as,
W = QN - Qout = 264.7 - 179.1 = 85.6 kJ/kg.
QA = W + Q3 - Q2
where Q3 = h3 and Q2 = (m2a * h2a + m2b * h2b + m2c * h2c)
Using these values, we get, QA = 85.6 + 191.81 - (0.0237 * 3053.7 + 0.0294 * 2987.2 + 0.0462 * 2834.9) = -56.5 kJ/kg.
(c) For an ideal engine and the same states, compute (d) W, QA, and e
The values for the ideal cycle can be calculated using the formulae,
e = 1 - (P3 / P1) ^ (γ - 1) / γ = 1 - (0.01 / 3.8) ^ 0.286 = 0.4821.
W = m (h1 - h3) = 0.0993 (3137.9 - 191.81) = 296.54 kJ/kg.
Qout = m (h3 - h4a) = 0.0993 (191.81 - 2873.2) = -266.96 kJ/kg.
QN = m (h1 - h4a) = 0.0993 (3137.9 - 2873.2) = 264.7 kJ/kg.
QA = W + Q3 - Q2
where Q3 = h3 and Q2 = 0,
Using these values, we get,QA = 296.54 + 191.81 = 488.35 kJ/kg
In conclusion, the given parameters were used to find the values for the amount of steam extracted, W, QA, and e for the ideal and regenerative Rankine cycle. The problem can be solved using the formulae provided and the enthalpy values from the steam table.
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A cylindrical specimen of some metal alloy 10 mm in diameter is stressed elastically in tension. A force of 10,000 N produces a reduction in specimen diameter of 2 × 10^-3 mm. The elastic modulus of this material is 100 GPa and its yield strength is 100 MPa. What is the Poisson's ratio of this material?
A cylindrical specimen of some metal alloy 10 mm in diameter is stressed elastically in tension.A force of 10,000 N produces a reduction in specimen diameter of 2 × 10^-3 mm.
The elastic modulus of this material is 100 GPa and its yield strength is 100 MPa.Poisson’s ratio (v) is equal to the negative ratio of the transverse strain to the axial strain. Mathematically,v = - (delta D/ D) / (delta L/ L)where delta D is the diameter reduction and D is the original diameter, and delta L is the length elongation and L is the original length We know that; Diameter reduction = 2 × 10^-3 mm = 2 × 10^-6 mL is the original length => L = πD = π × 10 = 31.42 mm.
The axial strain = delta L / L = 0.0032/31.42 = 0.000102 m= 102 μm Elastic modulus (E) = 100 GPa = 100 × 10^3 M PaYield strength (σy) = 100 MPaThe stress produced by the force is given byσ = F/A where F is the force and A is the cross-sectional area of the specimen. A = πD²/4 = π × 10²/4 = 78.54 mm²σ = 10,000/78.54 = 127.28 M PaSince the stress is less than the yield strength, the deformation is elastic. Poisson's ratio can now be calculated.v = - (delta D/ D) / (delta L/ L)= - 2 × 10^-6 / 10 / (102 × 10^-6) = - 0.196Therefore, the Poisson's ratio of this material is -0.196.
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Polyethylene (PE), C2H4 has an average molecular weight of 25,000 amu. What is the degree of polymerization of the average PE molecule? Answer must be to 3 significant figures or will be marked wrong. Atomic mass of Carbon is 12.01 Synthesis is defined as a. The shaping of materials into components to cause changes in the properties of materials.
b. The making of a material from naturally occurring and/or man-made material. c. The arrangement and rearrangement of atoms to change the performance of materials. d. The chemical make-up of naturally occurring and/or engineered material.
The degree of polymerization (DP) of a polymer is defined as the average number of monomer units in a polymer chain.the degree of polymerization of the average PE molecule is approximately 890.
In the case of polyethylene (PE), which has an average molecular weight of 25,000 amu, we can calculate the DP using the formula:
DP = (Average molecular weight of polymer) / (Molecular weight of monomer)
The molecular weight of ethylene (C2H4) can be calculated as follows:
Molecular weight of C2H4 = (2 * Atomic mass of Carbon) + (4 * Atomic mass of Hydrogen)
= (2 * 12.01 amu) + (4 * 1.01 amu)
= 24.02 amu + 4.04 amu
= 28.06 amu
Now, we can calculate the DP:
DP = 25,000 amu / 28.06 amu
≈ 890.24
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Two of the six stages are not pulling any current. It looks like a sequencer problem at first. A voltage check of the individual sequencer coils shows that all of the sequencers should have their contacts closed; there is 24 V at each coil. A voltage check at each heater terminal shows that all stages have voltage but are not drawing any current.
What is the likely problem and the recommended solution?
Based on the provided information, the likely problem is a malfunctioning sequencer coil, specifically the contacts that are not closing despite receiving the proper voltage.
This is causing the stages to have voltage but not draw any current. The sequencer is responsible for controlling the activation of the heating elements in each stage, so if the contacts fail to close, the heating elements won't receive power.
The recommended solution is to replace the faulty sequencer coil. Since all the sequencer coils are receiving the correct voltage, it indicates that the power supply and wiring are functioning correctly.
However, the contacts within the problematic sequencer coil are likely worn out or damaged, preventing them from closing properly.
To fix the issue, you should acquire a new sequencer coil that matches the specifications of the existing one. Turn off the power to the system before proceeding.
Remove the cover of the sequencer compartment and locate the faulty coil. Disconnect the electrical connections and remove the defective coil from its mounting.
Install the new sequencer coil in its place, ensuring proper alignment and connection of the electrical terminals. Finally, replace the cover and restore power to the system.
It is essential to consult the equipment's manual or contact a professional technician familiar with the specific system to ensure safe and accurate troubleshooting and repair.
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Dodecane (C12H26) can be used as a good approximation to Diesel fuel. a) Determine the gravimetric air-to-fuel ratio for the complete combustion of dodecane in air. [5 marks] b) Low temperature combustion is being developed in Diesel engines to reduce the NOX emissions. Discuss why NO emissions in Diesel engines are generally higher compared to Gasoline enginer and how this technology will reduce the NO_x.
(a) The gravimetric air-to-fuel ratio for the complete combustion of dodecane in air needs to be determined. (b) Diesel engines generally have higher NO emissions compared to gasoline engines.
(a) To determine the gravimetric air-to-fuel ratio for the complete combustion of dodecane in air, we need to consider the stoichiometric ratio. For complete combustion, the ideal air-to-fuel ratio provides sufficient oxygen for the complete oxidation of the fuel. By balancing the chemical equation for the combustion of dodecane (C12H26 + 18.5O2 → 12CO2 + 13H2O), we find that 18.5 moles of oxygen are required for 1 mole of dodecane. From the molecular weights, we can convert these moles to grams and determine the corresponding weight ratio of air to dodecane. (b) Diesel engines tend to have higher NO emissions compared to gasoline engines due to the higher combustion temperatures.
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Load has a power factor of — (lagging) 1) is this lood inductive or capacitive 2) You connect a test voltage source of 20my (cosωt) and measure the current. Absolute value of the current is 5mA. Find its phase. and 3) Find average power, reactive power apparent power Draw the vectors.
The real power, or average power, is represented by the adjacent side of the triangle, while the reactive power is represented by the opposite side. The real power vector is horizontal, while the reactive power vector is vertical.
Load has a power factor of 1, which is lagging, indicating that the load is inductive. The load is inductive because the power factor is lagging and is between 0 and 1. A lagging power factor indicates that the current is not in phase with the voltage.
The test voltage source is 20mV (cosωt), and the absolute value of the current is 5mA. To determine the phase angle, we'll need to use Ohm's law.
Since the current and voltage are out of phase, we'll need to utilize complex arithmetic to determine the phase angle. We'll have to compute the product of the two complex numbers.
In this case, Z=V/I,
where V = 20mV,
I = 5mA.
Therefore, Z = (20 x 10^-3)/(5 x 10^-3) = 4.
The angle of this complex number is the same as the phase angle of the circuit.
Therefore, tan θ = 0.5, and θ = 26.56 degrees.
The following formulae were used to find the average power, reactive power, and apparent power:
Average power = Vrms * Irms * cosθ = 20mV * 5mA * cos 26.56 degrees
= 0.444mWReactive power
= Vrms * Irms * sinθ
= 20mV * 5mA * sin 26.56 degrees
= 0.208mWApparent power
= Vrms * Irms = 20mV * 5mA
= 0.1mW
The power vectors can be drawn to represent the power characteristics of the circuit. The apparent power is represented by the hypotenuse of the power triangle.
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Explain briefly the advantages" and "disadvantages of the "Non ferrous metals and alloys" in comparison with the "Ferrous alloys (15p). Explain briefly the compositions and the application areas of the "Brasses"
The advantages are : 1. Non-ferrous metals are generally more corrosion resistant than ferrous alloys. 2. They are also more lightweight and have a higher melting point. 3. Some non-ferrous metals, such as copper, are excellent conductors of electricity. The disadvantages are : 1. Non-ferrous metals are typically more expensive than ferrous alloys. 2. They are also more difficult to machine and weld. 3. Some non-ferrous metals, such as lead, are toxic.
Here is a brief explanation of the compositions and application areas of brasses:
1. Brasses are copper-based alloys that contain zinc.
2. The amount of zinc in a brass can vary, and this can affect the properties of the alloy.
3. For example, brasses with a high zinc content are more ductile and machinable, while brasses with a low zinc content are more resistant to corrosion.
4. Brasses are used in a wide variety of applications, including:
Electrical connectors
Plumbing fixtures
Musical instruments
Jewelry
Coins
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a) The Mechanical Efficiency of a machine is given as € = i) Power output/Power input. ii) Energy input/ Energy output iii) Power input/ Power output. iv) Energy output/ Energy input. only i; only ii; i and iv; ili and iv
Oiv OII. only i Olll. ii and it OIV. i and iv The efficiency of a machine is always equal to or greater than 1 (True/ False) The displacement of the particle is defined as the change in its position (True/False)
a) The Mechanical Efficiency of a machine is given as € = i) Power output/Power input. ii) Energy input/ Energy output iii) Power input/ Power output. iv) Energy output/ Energy input. only i; only ii; i and iv; ili and iv.The answer is i) Power output/Power input.
It is because the formula of mechanical efficiency of a machine is given as -Power output/ Power input. This formula is used to calculate the efficiency of a machine. It is the ratio of output power to input power of a machine. It represents how much of the input energy is converted into output energy. It is expressed as a percentage or decimal value. It can never be greater than 1.The efficiency of a machine is always equal to or greater than 1 (True/ False)The efficiency of a machine can never be greater than 1.
It can be equal to 1 or less than 1. An ideal machine has a 100% efficiency, so its efficiency will be equal to 1. The actual efficiency of a machine is always less than the ideal efficiency. Hence, the given statement is false.The displacement of the particle is defined as the change in its position (True/False)The given statement is true. Displacement is defined as the change in the position of an object or particle in a particular direction. It is a vector quantity, which means it has a magnitude as well as a direction. It is measured in meters (m) or any other unit of length. It is calculated by subtracting the initial position of the particle from the final position of the particle.
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List three (3) basic attributes required for the operation of PV Cells.
What technology is used to generate electricity from solar power?
Three basic attributes required for the operation of PV cells (Photovoltaic cells) are: Sunlight: PV cells require sunlight or solar radiation to generate electricity.
Semiconductor Material: PV cells are made of semiconductor materials, typically silicon-based, that have the ability to convert sunlight into electricity. Electric Field: PV cells have an internal electric field created by the junction between different types of semiconductor materials. This electric field helps separate the generated electron-hole pairs, allowing the flow of electric current.
The technology used to generate electricity from solar power is called solar photovoltaic technology or solar PV technology. Solar PV technology involves the use of PV cells to directly convert sunlight into electricity.This electric current can then be harnessed and used to power electrical devices or stored in batteries for later use.
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A 18.65 kW,4-pole, 50 Hz,3 phase induction motor has the friction and windage losses of 2.5% of the output power at rated load. If the stator copper loss and iron loss are 1.5% of the output power and the slip is 4% at rated load, evaluate i. the rotor copper loss; ii. the power transferred from stator to rotor; iii. the output torque; iv. the gross electromagnetic torque; and v. the efficiency.
i. The rotor copper loss = 0.014116 kW (or 14.116 W)
ii. The power transferred from stator to rotor = 16.477884 kW
iii. The output torque (T) = 0.03333 Nm
iv. The gross electromagnetic torque (Te) = 7.00987 Nm
v. The efficiency (η) = 95.4%
Given data:
Rated power (output power) = 18.65 kW
Friction and windage losses = 2.5% of the output power
Stator copper loss = 1.5% of the output power
Iron loss = 1.5% of the output power
Slip at rated load = 4%
Step 1: Calculate the rotor copper loss.
Rotor copper loss = Output power × slip × (stator copper loss + iron loss)
Rotor copper loss = 18.65 kW × 0.04 × (0.015 + 0.015) = 0.014116 kW (or 14.116 W)
Step 2: Calculate the power transferred from stator to rotor.
Power transferred from stator to rotor = Output power - (friction and windage losses + stator copper loss + iron loss + rotor copper loss)
Power transferred from stator to rotor = 18.65 kW - (0.025 × 18.65 kW + 0.015 × 18.65 kW + 0.015 × 18.65 kW + 0.014116 kW) = 16.477884 kW
Step 3: Calculate the output torque.
The output power of a 3-phase induction motor can be related to the output torque (T) and the synchronous speed (Ns) using the formula:
Output power = (3 × Vph × Iph × pf × η) / (2 × π × Ns)
Rearranging the formula to find the output torque:
Output torque (T) = (Output power × (2 × π × Ns)) / (3 × Vph × Iph × pf × η)
Assuming:
Vph = 400 V (phase voltage)
Iph = 25 A (phase current)
pf = 0.8 (power factor)
η = Efficiency (to be calculated)
Output torque (T) = (18.65 kW × (2 × π × 1500)) / (3 × 400 V × 25 A × 0.8 × η)
The output power of a 3-phase induction motor can be related to the output torque (T) and the synchronous speed (Ns) using the formula:
Output power = (3 × Vph × Iph × pf × η) / (2 × π × Ns)
Rearranging the formula to find the output torque:
Output torque (T) = (Output power × (2 × π × Ns)) / (3 × Vph × Iph × pf × η)
Assuming:
Vph = 400 V (phase voltage)
Iph = 25 A (phase current)
pf = 0.8 (power factor)
η = 95.4% (efficiency)
Output torque (T) = (18.65 kW × (2 × π × 1500)) / (3 × 400 V × 25 A × 0.8 × 0.954)
Calculating the value:
Output torque (T) = 0.03333 Nm
Therefore, the output torque is approximately 0.03333 Nm.
Step 4: Calculate the gross electromagnetic torque.
The gross electromagnetic torque (Te) can be calculated using the formula:
Te = (Power transferred from stator to rotor × 1000) / (2 × π × Ns)
Te = (16.477884 kW × 1000) / (2 × π × 1500) = 7.00987 Nm
Step 5: Calculate the efficiency.
Efficiency (η) = (Output power / Input power) × 100
Input power = Output power + losses
Losses = friction and windage losses + stator copper loss + iron loss + rotor copper loss
Losses = 0.025 × 18.65 kW + 0.015 × 18.65 kW + 0.015 × 18.65 kW + 0.014116 kW = 0.918375 kW
Input power = 18.65 kW + 0.918375 kW = 19.568375 kW
Efficiency (η) = (18.65 kW / 19.568375 kW) × 100 = 95.4%
Summary of Results:
i. The rotor copper loss = 0.014116 kW (or 14.116 W)
ii. The power transferred from stator to rotor = 16.477884 kW
iii. The output torque (T) = 0.03333 Nm
iv. The gross electromagnetic torque (Te) = 7.00987 Nm
v. The efficiency (η) = 95.4%
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A cantilever beam has length 24 in and a force of 2000 lbf at the free end. The material is A36/. For a factor of safety of 2, find the required cross section dimensions of the beam. The cross section can be assumed as square, rectangular, pipe or I-beam.
The formula for the shear stress in a cantilever beam subjected to a transverse force can be used to find the required cross-section dimensions for the beam.The formula is; τmax = VQ/ItWhere;V = the maximum force (2000 lbs.)Q = the first moment of the area around the neutral axis.
I = the moment of inertia.The maximum shear stress for A36 steel is 20,000 psi. For a factor of safety of 2, this value can be doubled to 40,000 psi.So,τmax = VQ/It = 40000 psi.The dimensions of the beam can be found using the shear stress equation and the bending moment equation.
Mmax = PL/4 = 2000 lbs. × 24 in./4 = 12000 in. lbs.τmax = Mmax*c/I = 40000 psiThe required cross-section dimensions of the beam can be found as follows;For a square beam;a = b ⇒ c = a / √6P = 12000 lbs.
[tex]Q = b × h × h / 2 = a × a × a / 2√3h = a/√3I = a^4/12c = I × τmax / b × h²a = (6 × P / (τmax × h²))^(1/4).[/tex]
For a rectangular beam;
[tex]a < b ⇒ c = a / √6P = 12000 lbs.Q = b × h × h / 2 = a × b × b / 2h = √(2a / 3)I = ab^3/12c = I × τmax / b × h²a = (6 × P / (τmax × h² × b))^(1/3) × b^2/3.[/tex]
For a pipe;a = b and D = 2rP = 12000 lbs.τavg = P/ (2A - a²) = 40000 psiThe diameter of the pipe can be found using the following equation;
[tex]r = (P/2τavg)(D² - d²)/D²d = D - 2ta = πr² - πr²/4A = πr²D = 2r(1 + (4a²/(πr^2))^(1/2)).[/tex]
For an I-beam;the required dimensions can be found by assuming that the beam is an equivalent rectangular beam and then using the above rectangular beam formula. In the equivalent rectangular beam, the width of the flanges is equal to the thickness of the web multiplied by a factor of 1.2 to 1.5. The thickness of the web is taken as the distance between the midpoints of the flanges.
From the above, we can conclude that the cross-section dimensions of a square beam, rectangular beam, pipe, and I-beam can be found.
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A closed system contains an ideal gas, which molecular weight is W-60 kg/kmol, and its standard state entropy is so-0. The system undergoes the following cycle: at state 1 the temperature is 293.15 K, the pressure is 94 kPa, and the entropy is 45.428 J/(kg.K). The gas is compressed polytropically at n=1.45 until the specific volume is 10 times lower than that at state 1 (state 2). Then 84552.2 J/kg of heat is added at constant specific volume (state 3). After that heat is added at constant pressure until entropy is 333.333 J/(kg.K) (state 4). In the next process the system undergoes isentropic expansion (and reaches state 5). Finally there is a constant volume rejection of heat (until state 1). Determine a) the values of p, v, T and s, at each cycle point
The constant volume rejection of heat from state 5 to state 1 means that the pressure and maxium temperature change, but the volume remains constant.
a) The values of **p, v, T, and s at each cycle point are as follows:
State 1:
p1 = 94 kPa
v1 = Unknown
T1 = 293.15 K
s1 = 45.428 J/(kg·K)
State 2:
p2 = Unknown
v2 = 10 * v1
T2 = Unknown
s2 = Unknown
State 3:
p3 = p2 (constant specific volume)
v3 = v2
T3 = Unknown
s3 = Unknown
State 4:
p4 = Unknown
v4 = Unknown
T4 = Unknown
s4 = 333.333 J/(kg·K)
State 5:
p5 = p1
v5 = Unknown
T5 = Unknown
s5 = s1
To determine the values at each state, we need to use the appropriate thermodynamic relationships and equations. The polytropic process in state 2 can be described using the equation p2 * v2^n = constant. The heat added at constant volume in state 3 does not affect the pressure, but increases the temperature. The heat added at constant pressure in state 4 increases the temperature and entropy.
The isentropic expansion from state 4 to state 5 implies that entropy remains constant. Finally, the constant volume rejection of heat from state 5 to state 1 means that the pressure and temperature change, but the volume remains constant. By applying the relevant equations and conditions, the values of p, v, T, and s at each state can be determined
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You are working as a Junior Engineer for a renewable energy consultancy. Your line manager is preparing a report for the local authority on the benefit of adopting renewable energy technology on their housing stock and civic buildings. You have been asked to contribute to the report by completing the following tasks, your work must be complete and accurate as it will be subject to scrutiny.
Activity
Tasks:
a) Determine the cost of installing a photo voltaic system on the roof of a two story house, it can be assumed that the roof is south facing. The available roof area is 4m x 4m, you will need to select suitable panels. Stating all assumptions estimate and detail the total cost of the installation and connection, then express this cost in terms of installed capacity (£/kW), this is known as the levelised cost.
Renewable energy systems are gaining popularity due to the benefits they offer. The cost of installing a photovoltaic system on the roof of a two-story house with a 4m x 4m south-facing roof will be determined in this article.
The levelized cost will be stated, which is the cost per installed capacity (£/kW).PV modules, inverters, racking equipment, and installation are the four components of a photovoltaic system. The cost of photovoltaic panels varies based on their size, wattage, and efficiency. The cost of photovoltaic panels is roughly £140-£180 per panel for 300W to 370W photovoltaic panels. A photovoltaic panel can generate 1 kW of electricity per day in good conditions.
It costs between £500 and £1000. Racking equipment will cost approximately £500, depending on the design and layout.Total installation cost:PV panels cost: 10 panels × £140 - £180 = £1400 - £1800Inverter cost: £500 - £1000Racking equipment cost: £500Installation cost: £1200 - £2000Total installation cost: £3600 - £5300Levelized cost: Levelized cost expresses the cost of the installation and connection in terms of installed capacity (£/kW). Installed capacity can be calculated by dividing the total PV panel capacity by 1,000.
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Calculate the jet velocity in m/s we would expect from the ideal ramjet at Mach = 4, at Tatm = 216.65 K and Patm = 7505 Pa if the combustion chamber is at T₀=2400 K and f = 2.1213. Use Cp=1005 kg⁻¹ k⁻¹.
The ideal jet velocity produced by the Ramjet engine is 1984.58 m/s (approximately). A Ramjet is an engine that produces thrust directly from oxygen in the air that passes through it.
The velocity of the jet produced from the ideal Ramjet at Mach = 4, at Tatm = 216.65 K and Patm = 7505 Pa is to be calculated, given that the combustion chamber is at T₀=2400 K and f = 2.1213.The formula for calculating the ideal jet velocity in a Ramjet engine is given by:
[tex]vj=√2CpT₀(1−(Patm/P₀)^((γ−1)/γ))[/tex]
T₀ is the temperature at the combustion chamber Patm is the atmospheric pressureγ is the ratio of specific heats
P₀ is the pressure at the combustion chamber (Pa )Substituting the given values in the above equation,
[tex]vj=√2×1005×2400×(1−(7505/101325)^((1.4−1)/1.4))=1984.58 m/s[/tex]
The ideal jet velocity produced by the Ramjet engine is 1984.58 m/s (approximately).
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