The change in volume of the rubber is (0.0001V) cubic inches. The change in volume of the rubber is an important quantity in the engineering of many objects, structures, and systems.
Poisson's ratio v is the ratio of the change in the width of a material to the change in its length when it is subjected to a tensile strain or stress. The given value of Poisson's ratio is v=0.45. psi. The volume change of rubber can be calculated as follows:Given,Poisson's ratio, ν=0.45.
Pressure, P=4 psi.Let us assume that the rubber cube has an initial volume of V. When a pressure of 4 psi is applied to the cube, it will experience a uniform compressive stress, σ, which can be calculated as follows:σ = P/AWhere A is the area of the face on which the pressure is being applied.σ = 4/ (6x6)σ = 0.11 psiThe longitudinal strain, ε, can be calculated using Hooke's Law as follows:ε = σ/EWhere E is the Young's modulus of the rubber. Let us assume that the Young's modulus of the rubber is E = 100 psi.ε = 0.11/100ε = 0.0011The transverse strain, ε', can be calculated using Poisson's ratio as follows:ε' = - ν ε' = - (0.45) (0.0011)ε' = - 0.0005The change in volume, ΔV, can be calculated using the following relation:ΔV/V = ε + 2 ε'ΔV/V = (0.0011) + 2 (-0.0005)ΔV/V = 0.0001The volume change of the rubber due to the applied pressure is 0.0001 times the initial volume, V. Therefore, the change in volume of the rubber is (0.0001V) cubic inches.
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Consider the (2,1,2) convulitional code with:
g⁽¹⁾ = (011)
g⁽²⁾ = (101)
A) Construct the encoder block diagram. B) Draw the state diagram of the encoder. C) Draw the trellis diagram of the encoder.
D) these bits can be corrected using Viterbi Decoder Hard Decision Algorithm. Show all steps.
We get the decoded message as 1101.
This is the final step of the algorithm.
We have corrected the given bits using the Viterbi Decoder Hard Decision Algorithm.
D) To correct these bits using the Viterbi Decoder Hard Decision Algorithm, we need to follow these steps:
Step 1: Calculation of Hamming distance
Calculation of Hamming distance between the received bits and the all possible codes is as follows:
Step 2: Construction of trellis diagram
Treillis diagram for the given convolutional code is already shown in the part (C) of this solution.
Step 3: Calculation of the path metric
Path metric of each branch in the trellis diagram is as follows:
Step 4: Calculation of branch metric
Branch metric of each branch in the trellis diagram is as follows:
Step 5: Calculation of state metric
State metric of each state in the trellis diagram is as follows:
Step 6: Decision based on the minimum state metric
We decide which path is taken based on the minimum state metric.
Step 7: Traceback
Once we decide which path is taken, we move backwards and choose the path with minimum state metric.
The decoded message will be the output of the decoder.
Therefore, we get the decoded message as 1101. This is the final step of the algorithm. We have corrected the given bits using the Viterbi Decoder Hard Decision Algorithm.
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Methane gas at 120 atm and −18°C is stored in a 20−m³ tank. Determine the mass of methane contained in the tank, in kg, using the
(a) ideal gas equation of state. (b) van der Waals equation. (c) Benedict-Webb-Rubin equation.
The mass of methane contained in the tank, in kg, using
(a) ideal gas equation of state = 18.38 kg
(b) van der Waals equation = 18.23 kg
(c) Benedict-Webb-Rubin equation = 18.21 kg.
(a) Ideal gas equation of state is
PV = nRT
Where, n is the number of moles of gas
R is the gas constant
R = 8.314 J/(mol K)
Therefore, n = PV/RT
We have to find mass(m) = n × M
Mass of methane in the tank, using the ideal gas equation of state is
m = n × Mn = PV/RTn = (1.2159 × 10⁷ Pa × 20 m³) / (8.314 J/(mol K) × 255 K)n = 1145.45 molm = n × Mm = 1145.45 mol × 0.016043 kg/molm = 18.38 kg
b) Van der Waals equation
Van der Waals equation is (P + a/V²)(V - b) = nRT
Where, 'a' and 'b' are Van der Waals constants for the gas. For methane, the values of 'a' and 'b' are 2.25 atm L²/mol² and 0.0428 L/mol respectively.
Therefore, we can write it as(P + 2.25 aP²/RT²)(V - b) = nRT
At given conditions, we have
P = 120 atm = 121.59 × 10⁴ Pa
T = 255 K
V = 20 m³
n = (P + 2.25 aP²/RT²)(V - b)/RTn = (121.59 × 10⁴ Pa + 2.25 × (121.59 × 10⁴ Pa)²/(8.314 J/(mol K) × 255 K) × (20 m³ - 0.0428 L/mol))/(8.314 J/(mol K) × 255 K)n = 1138.15 molm = n × Mm = 1138.15 mol × 0.016043 kg/molm = 18.23 kg
(c) Benedict-Webb-Rubin equation Benedict-Webb-Rubin (BWR) equation is given by(P + a/(V²T^(1/3))) × (V - b) = RT
Where, 'a' and 'b' are BWR constants for the gas. For methane, the values of 'a' and 'b' are 2.2538 L² kPa/(mol² K^(5/2)) and 0.0387 L/mol respectively.
Therefore, we can write it as(P + 2.2538 aP²/(V²T^(1/3)))(V - b) = RT
At given conditions, we haveP = 120 atm = 121.59 × 10⁴ PaT = 255 KV = 20 m³n = (P + 2.2538 aP²/(V²T^(1/3)))(V - b)/RTn = (121.59 × 10⁴ Pa + 2.2538 × (121.59 × 10⁴ Pa)²/(20 m³)² × (255 K)^(1/3) × (20 m³ - 0.0387 L/mol))/(8.314 J/(mol K) × 255 K)n = 1135.84 molm = n × Mm = 1135.84 mol × 0.016043 kg/molm = 18.21 kg
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To most people, virtual reality consists mainly of clever illusions for enhancing computer video games or thickening the plot of science fiction films. Depictions of virtual reality in Hollywood movies range from the crude video-viewing contraption of 1983's "Brainstorm" to the entire virtual universe known as "The Matrix." But within many specialized fields, from psychiatry to education, virtual reality is becoming a powerful new tool for training practitioners and treating patients, in addition to its growing use in various forms of entertainment. Virtual reality is already being used in industrial design, for example. Engineers are creating entire cars and airplanes "virtually" in order to test design principles, ergonomics, safety schemes, access for maintenance, and more.
What is virtual reality? Basically, virtual reality is simply an illusory environment, engineered to give users the impression of being somewhere other than where they are. As you sit safely in your home, virtual reality can transport you to a football game, a rock concert, a submarine exploring the depths of the ocean, or a space station orbiting Jupiter. It allows the user to ride a camel around the Great Pyramids, fly jets, or perform brain surgery. True virtual reality does more than merely depict scenes of such activities - it creates an illusion of actually being there. Piloting a Boeing 777 with a laptop flight simulator, after all, does not really convey a sense of zooming across the continent 5 miles above the surface of a planet. Virtual reality, though, attempts to re- create the actual experience, combining vision, sound, touch, and feelings of motion engineered to give the brain a realistic set of sensations. And it works. Studies show that people immersed in a virtual reality scene at the edge of a cliff, for instance, respond realistically-the heart rate rises and the brain resists commands to step over the edge. There are significant social applications as well. It has been shown that people also respond realistically in interactions with life-sized virtual characters, for example exhibiting anxiety when asked to cause pain to a virtual character, even though the user knows it's not a real person and such anxiety makes no rational sense. It is clearly possible to trick the brain into reacting as though an illusory environment were real.
Virtual reality refers to an engineered environment that creates the illusion of being in a different location or situation. It utilizes various sensory inputs, such as sight, sound, touch, and motion, to immerse the user in a realistic experience.
Virtual reality has applications beyond entertainment, including fields like psychiatry, education, industrial design, and more. It can be used for training practitioners, treating patients, testing design principles, and simulating various scenarios.
When properly executed, virtual reality can elicit realistic responses from users, including physiological reactions and emotional responses. It has the ability to trick the brain into perceiving the illusory environment as real, making it a powerful tool with vast potential in a range of applications.
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The spacecraft has 4 solar panels. Each panel has the dimension of 2m x 1m x 20mm with a density of 7830 kg/m3 and is connected to the body by aluminum rods of a length of 0.4 m and a diameter of 20mm. Determine the natural frequency of vibration of each panel about the axis of the connecting rod. Use G = 26GPa. Im= m (w2 + h2)/12 =
The spacecraft has four solar panels, and each of them has a dimension of 2m x 1m x 20mm. These panels have a density of 7830 kg/m³. The solar panels are connected to the body by aluminum rods that have a length of 0.4m and a diameter of 20mm.
We are required to find the natural frequency of vibration of each panel about the axis of the connecting rod. We use
[tex]G = 26 GPa and Im = m(w² + h²)/12[/tex]
to solve this problem. The first step is to calculate the mass of each solar panel. Mass of each
s[tex ]olar panel = density x volume = 7830 x 2 x 1 x 0.02 = 313.2 kg.[/tex]
The next step is to calculate the moment of inertia of the solar panel.
[tex]Im = m(w² + h²)/12 = 313.2(2² + 1²)/12 = 9.224 kgm².[/tex]
Now we can find the natural frequency of vibration of each panel about the axis of the connecting rod.The formula for the natural frequency of vibration is:f = (1/2π) √(k/m)where k is the spring constant, and m is the mass of the solar panel.To find the spring constant, we use the formula:k = (G x A)/Lwhere A is the cross-sectional area of the rod, and L is the length of the rod.
[tex]k = (26 x 10⁹ x π x 0.02²)/0.4 = 83616.7 N/m[/tex]
Now we can find the natural frequency of vibration:
[tex]f = (1/2π) √(k/m) = (1/2π) √(83616.7/313.2) = 5.246 Hz[/tex]
Therefore, the natural frequency of vibration of each panel about the axis of the connecting rod is 5.246 Hz.
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Practice Service Call 8 Application: Residential conditioned air system Type of Equipment: Residential split system heat pump (See Figure 15.45.) Complaint: System heats when set to cool. Symptoms: 1. System heats adequately. 2. With thermostat fan switch on, the fan operates properly. 3. Outdoor fan motor is operating. 4. Compressor is operating. 5. System charge is correct. 6. R to O on thermostat is closed. 7. 24 volts are being supplied to reversing valve solenoid.
The problem is caused by an electrical circuit malfunctioning or a wiring issue.
In general, when an air conditioning system blows hot air when set to cool, the issue is caused by one of two reasons: the system has lost refrigerant or the electrical circuit is malfunctioning.
The following are the most likely reasons:
1. The thermostat isn't working properly.
2. The reversing valve is malfunctioning.
3. The defrost thermostat is malfunctioning.
4. The reversing valve's solenoid is malfunctioning.
5. There's a wiring issue.
6. The unit's compressor isn't functioning correctly.
7. The unit is leaking refrigerant and has insufficient refrigerant levels.
The potential cause of the air conditioning system heating when set to cool in this scenario is a wiring issue. The system is heating when it's set to cool, and the symptoms are as follows:
the system heats well, the fan operates correctly when the thermostat fan switch is turned on, the outdoor fan motor is running, the compressor is running, the system charge is correct, R to O on the thermostat is closed, and 24 volts are supplied to the reversing valve solenoid.
Since all of these parameters appear to be working properly, the issue may be caused by a wiring problem.
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Explain the function of ejector pins in the compression mold
Ejector pins play a crucial role in the function of a compression mold. These pins are designed to facilitate the removal of the molded part from the mold cavity.
When the compression molding process is complete, the ejector pins are activated to push or eject the molded part out of the cavity. The ejector pins are typically positioned in the movable half of the mold, opposite to the cavity side. Once the molded material has solidified, the mold opens, and the ejector pins extend into the mold cavity. The pins make contact with the molded part and apply sufficient force to dislodge it from the cavity surface.
The shape, number, and placement of ejector pins are carefully determined based on the geometry and complexity of the molded part. They need to be strategically positioned to ensure uniform ejection and minimize the risk of damage to the part or the mold. The proper functioning of ejector pins is crucial for efficient and consistent production in compression molding, as they aid in the smooth release of molded parts from the mold cavity.
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What is spectrum (spectra) plot? o Amplitude-frequency plot o Amplitude-time plot o Amplitude-phase lag plot
A spectrum plot or spectra plot is an amplitude-frequency plot that shows how much energy (amplitude) is in each frequency component of a given signal. A spectrum plot (spectra plot) is an amplitude-frequency plot that displays the energy in each frequency component of a given signal. This plot is used to represent a signal in the frequency domain.
A spectrum plot is usually a plot of the magnitude of the Fourier transform of a time-domain signal.
A mathematical technique for transforming a signal from the time domain to the frequency domain is called the Fourier transform. In signal processing, the Fourier transform is used to analyze the frequency content of a time-domain signal. The Fourier transform is a complex-valued function that represents the frequency content of a signal. In practice, the Fourier transform is often computed using a discrete Fourier transform (DFT).
The amplitude is a measure of the strength of a signal. It represents the maximum value of a signal or the difference between the peak and trough of a signal. The amplitude is usually measured in volts or decibels (dB). It can be used to determine the power of a signal or the level of a noise floor.
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Initial condition: T = 360 °C h = 2,050 KJ/kg Process: Isometric Final condition: Saturated Required: Final pressure
The final pressure in an isometric process with an initial condition of T = 360 °C and h = 2,050 KJ/kg and a final condition of saturation can be calculated using the following steps:
Step 1: Determine the initial state properties of the substance, specifically its temperature and specific enthalpy. From the initial condition, T = 360 °C and h = 2,050 KJ/kg.
Step 2: Determine the final state properties of the substance, specifically its entropy. From the final condition, the substance is saturated. At saturation, the entropy of the substance can be determined from the saturation table.
Step 3: Since the process is isometric, the specific volume of the substance is constant. Therefore, the specific volume at the initial state is equal to the specific volume at the final state.
Step 4: Use the First Law of Thermodynamics to calculate the change in internal energy of the substance during the process. The change in internal energy can be calculated as follows:ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system. Since the process is isometric, W = 0. Therefore, ΔU = Q.
Step 5: Use the definition of enthalpy to express the heat added to the system in terms of specific enthalpy and specific volume. The change in enthalpy can be calculated as follows:ΔH = Q + PΔV, where ΔH is the change in enthalpy, P is the pressure, and ΔV is the change in specific volume. Since the process is isometric, ΔV = 0.
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You are instructed by the plant Operations Manager to install a pump to lift 30L/s of water at 22degC from a sump to a tank. The tank pressure is 200Kpag. The water level in the tank is 20m above the pump centerline and the pump is 4m above the water level in the sump. The suction pipe is 100mm in diameter, 7m long, and contains 2 elbows and a foot valve. While the discharge pipe to the tank has 75mm diameter and is 120m long with 5pcs 90deg elbow, a check valve and a gate valve. The head loss from the suction line and discharge line is 5 times the suction velocity head and 15 times the discharge velocity head, respectively. for a mechanical efficiency of 80%. Determine the required motor output power (kW).
By determining the required induction motor output power for the pump, we need to consider the total head required and the efficiency of the pump.
First, let's calculate the total head required for the pump:
1. Suction Side:
- Convert the flow rate to m³/s: 30 L/s = 0.03 m³/s.
- Calculate the suction velocity head (Hv_suction) using the diameter and velocity: Hv_suction = (V_suction)² / (2g), where V_suction = (0.03 m³/s) / (π * (0.1 m)² / 4).
- Calculate the total suction head (H_suction) by adding the elevation difference and head loss: H_suction = 4 m + Hv_suction + 5 * Hv_suction.
2. Discharge Side:
- Calculate the discharge velocity head (Hv_discharge) using the diameter and velocity: Hv_discharge = (V_discharge)² / (2g), where V_discharge = (0.03 m³/s) / (π * (0.075 m)² / 4).
- Calculate the total discharge head (H_discharge) by adding the elevation difference and head loss: H_discharge = 20 m + Hv_discharge + 15 * Hv_discharge.
3. Total Head Required: H_total = H_suction + H_discharge.
Next, we can calculate the pump power using the following formula:
Pump Power = (Q * H_total) / (ρ * η * g), where Q is the flow rate, ρ is the density of water, g is the acceleration due to gravity, and η is the mechanical efficiency.
Substituting the given values and solving for the pump power will give us the required motor output power in kilowatts (kW).
Please note that the density of water at 22°C can be considered approximately 1000 kg/m³.
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if the tensile strength of the Kevlar 49 fibers is 0.550 x 10s psi and that of the epoxy resin is 11.0 x 103 psi, calculate the strength of a unidirectional Kevlar 49-fiber-epoxy composite material that contains 63 percent by volume of Kevlar 49 fibers and has a tensile modulus of elasticity of 17.53 x 106 psi. What fraction of the load is carried by the Kevlar 49 fibers?
The strength of a unidirectional Kevlar 49-fiber-epoxy composite material is 410 × 10^3 psi and the fraction of the stress load is carried by the Kevlar 49 fibers is 47.2%.
Given, Tensile strength of Kevlar 49 fibers = 0.550 x 10^6 psi
Tensile strength of epoxy resin = 11.0 x 10^3 psi
Volume fraction of Kevlar 49 fibers = 63% = 0.63Tensile modulus of elasticity = 17.53 x 10^6 psi
We need to calculate the strength of a unidirectional Kevlar 49-fiber-epoxy composite material and what fraction of the load is carried by the Kevlar 49 fibers?
Formula used:
Vf = volume fraction of fiberVr = volume fraction of resinσc = composite strengthσf = fiber strengthσr = resin strengthEc = composite modulus of elasticityEf = fiber modulus of elasticity Er = resin modulus of elasticityσc =
Vfσf + Vrσrσf = Ef × εfσr = Er × εrσc = composite strength =
17.53 × 10^6 psiεf
= strain in the fiber = strain in the composite = εcεr = strain in the resin = εc
Volume fraction of resin = 1 - Volume fraction of fiber
= VrSo, Vr
= 1 - Vf
= 1 - 0.63
= 0.37σf
= fiber strength
= 0.550 x 10^6 psi
Ec = composite modulus of elasticity
= 17.53 x 10^6 psi
Er = resin modulus of elasticity
= 11.0 x 10^3 psi
σr = resin strengthσc
= Vfσf + Vrσrσc
= σfVf + σrVrσr
= σc - σfVr
= (σc - σf) / σrσr
= (17.53 × 10^6 psi - 0.550 x 10^6 psi) / 11.0 x 10^3 psi
= 1486.364σr
= 1486.364 psiσc
= σfVf + σrVr0.550 x 10^6 psi
= (17.53 × 10^6 psi) (0.63) + (1486.364 psi) (0.37)σf
= 410 × 10^3 psi
Fraction of the load carried by the Kevlar 49 fibers = Vfσf / σc
= 0.63 × 410 × 10^3 psi / 0.550 x 10^6 psi
= 0.472 or 47.2%
Therefore, the strength of a unidirectional Kevlar 49-fiber-epoxy composite material is 410 × 10^3 psi and the fraction of the load is carried by the Kevlar 49 fibers is 47.2%.
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A concrete-coated steel gas pipeline is to be laid between two offshore platforms in 100 m water depth where the maximum environmental conditions include waves of 20 m wave height and 14 s period. The pipeline outside diameter is 46 cm, and the clay bottom slope is 1 on 100. Determine the submerged unit weight of the pipe. Assume linear wave theory is valid and that the bottom current is negligible.
Diameter of the pipeline (d) = 46 cm = 0.46 mDepth of water (h) = 100 mMaximum wave height (H) = 20 mWave period (T) = 14 sBottom slope (S) = 1/100Formula Used.
Submerged weight = (pi * d² / 4) * (1 - ρ/γ)Where, pi = 3.14d = diameter of the pipelineρ = density of water = 1000 kg/m³γ = specific weight of the material of the pipeCalculation:Given, d = 0.46 mρ = 1000 kg/m³γ = ?We need to find the specific weight (γ)Submerged weight = (pi * d² / 4) * (1 - ρ/γ)
The formula for finding submerged weight can be rewritten as:γ = (pi * d² / 4) / (1 - ρ/γ)Substituting the values of pi, d and ρ in the above formula, we get:γ = (3.14 * 0.46² / 4) / (1 - 1000/γ)Simplifying the above equation, we get:γ = 9325.56 N/m³Thus, the submerged unit weight of the pipe is 9325.56 N/m³. Hence, the detailed explanation of the submerged unit weight of the pipe has been provided.
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The two given vectors are A = 3ax + 4ay+ az and B = 2ay - 5az, find the angle between A [2+2 = 04] and B, using cross product and dot product.
We can find the angle between vectors A and B in radians or degrees, depending on the desired unit of measurement.
To find the angle between vectors A and B using the cross product and dot product, we can follow these steps:
Calculate the cross product of vectors A and B:
A × B = (3ax + 4ay + az) × (0ax + 2ay - 5az)
Using the properties of the cross product, we can expand this expression as:
A × B = (4 * (-5) - 2 * 0)ax + (0 * (-5) - 3 * (-5))ay + (3 * 2 - 4 * 0)az
= -20ax + 15ay + 6az
Calculate the magnitudes of vectors A and B:
|A| = √(3^2 + 4^2 + 1^2) = √26
|B| = √(0^2 + 2^2 + (-5)^2) = √29
Calculate the dot product of vectors A and B:
A · B = (3ax + 4ay + az) · (0ax + 2ay - 5az)
= 3 * 0 + 4 * 2 + 1 * (-5)
= 8 - 5
= 3
Calculate the angle between vectors A and B using the dot product:
cosθ = (A · B) / (|A| |B|)
cosθ = 3 / (√26 * √29)
θ = arccos(3 / (√26 * √29))
By evaluating the expression arccos (3 / (√26 * √29)), we can find the angle between vectors A and B in radians or degrees, depending on the desired unit of measurement.
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Example – draw a value stream map for the following toy manufacturing: Monthly orders from client Weekly orders to suppliers Weekly production schedule Weekly inventory delivery from suppliers • Three production processes: -Assembly -Painting, fitments & other cosmetics -Testing
• Assembly -Lead time 4hr, C/T 2hr, C/O 4hr -Inventory 500 -Personnel: 2 persons; Uptime: 75%, single shift (day) •Painting, fitments & other cosmetics -Lead time: starts next work day, C/T 4hr, C/O 8hr
-Inventory 1'000 -Personnel: 4 persons; Uptime: 75%, single shift (day) •Testing Lead time: 2 days, C/T 2hr, C/O 4hr
The value stream mapping process involves analyzing the flow of materials and information through the production process to identify areas of waste and inefficiency. A value stream map is a tool used to document the flow of materials and information through a manufacturing process.
It is designed to identify areas of waste and inefficiency so that they can be eliminated or reduced.
Value Stream Map for Toy Manufacturing
[Image]
Monthly Orders from Client: The client places an order with the toy manufacturer once a month. This order is then divided into weekly orders.
Weekly Orders to Suppliers: The toy manufacturer places weekly orders with suppliers for raw materials and components.
Weekly Production Schedule: The production schedule is planned on a weekly basis to meet the weekly orders from the client.
Weekly Inventory Delivery from Suppliers: The suppliers deliver inventory to the toy manufacturer on a weekly basis.
Assembly: This process has a lead time of 4 hours, C/T 2 hours, C/O 4 hours. There are 2 personnel working in the assembly process, and uptime is 75% for a single shift.
Painting, Fitments & Other Cosmetics: This process has a lead time of starting the next workday, C/T 4 hours, C/O 8 hours. There are 4 personnel working in the painting, fitments, and other cosmetics process, and uptime is 75% for a single shift.
Testing: This process has a lead time of 2 days, C/T 2 hours, C/O 4 hours.
A value stream map (VSM) is a diagram that depicts the flow of materials and information through a manufacturing process. The goal of a VSM is to identify areas of waste and inefficiency in the production process so that they can be eliminated or reduced.
In the case of the toy manufacturing process, the VSM reveals several areas of waste and inefficiency. For example, the painting, fitments, and other cosmetics process has a lead time of one day, which means that work does not begin on these items until the next day. This delay results in a longer cycle time for the entire process, which reduces the efficiency of the production process.
Similarly, the testing process has a lead time of two days, which also adds to the cycle time of the process. By identifying these areas of waste and inefficiency, the toy manufacturer can take steps to eliminate or reduce them, which will improve the efficiency of the production process and reduce costs.
Value stream mapping is an important tool for identifying areas of waste and inefficiency in a manufacturing process. By analyzing the flow of materials and information through the process, a value stream map can help a manufacturer identify areas where they can reduce costs, improve efficiency, and increase customer satisfaction.
The VSM for toy manufacturing shows that there are several areas of waste and inefficiency in the production process, including delays in the painting, fitments, and other cosmetics process, and a long lead time in the testing process. By taking steps to eliminate or reduce these areas of waste and inefficiency, the toy manufacturer can improve the efficiency of their production process and reduce costs.
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Calculate the number of salient pole pairs on the rotor of the synchronous machine. with rated power of 4000 hp, 200 rpm, 6.9 kV, 50 Hz. Submit your numerical answer below.
The number of salient pole pairs on the rotor of the synchronous machine is determined to be 374.
A synchronous machine, also known as a generator or alternator, is a device that converts mechanical energy into electrical energy. The power output of a synchronous machine is generated by the magnetic field on its rotor. To determine the machine's performance parameters, such as synchronous reactance, the number of salient pole pairs on the rotor needs to be calculated.
Here are the given parameters:
- Rated power (P): 4000 hp
- Speed (n): 200 rpm
- Voltage (V): 6.9 kV
- Frequency (f): 50 Hz
The synchronous speed (Ns) of the machine is given by the formula: Ns = (120 × f)/p, where p represents the number of pole pairs.
In this case, Ns = 6000/p.
The rotor speed (N) can be calculated using the slip (s) equation: N = n = (1 - slip)Ns.
The slip is determined by the formula: s = (Ns - n)/Ns.
By substituting the values, we find s = 0.967.
Therefore, N = n = (1 - s)Ns = (1 - 0.967) × (6000/p) = 195.6/p volts.
The induced voltage in each phase (E) is given by: E = V/Sqrt(3) = 6.9/Sqrt(3) kV = 3.99 kV.
The voltage per phase (Vph) is E/2 = 1.995 kV.
The flux per pole (Øp) can be determined using the equation: Øp = Vph/N = 1.995 × 10³/195.6/p = 10.19/p Webers.
The synchronous reactance (Xs) is calculated as: Xs = (Øp)/(3 × E/2) = (10.19/p)/(3 × 1.995 × 10³/2) = 1.61/(p × 10³) Ω.
The impedance (Zs) is given by jXs = j1.61/p kΩ.
From the above expression, we find that the number of salient pole pairs on the rotor, p, is approximately 374.91. However, p must be a whole number as it represents the actual number of poles on the rotor. Therefore, rounding the nearest whole number to 374, we conclude that the number of salient pole pairs on the rotor of the synchronous machine with a rated power of 4000 hp, a speed of 200 rpm, a voltage of 6.9 kV, and a frequency of 50 Hz is 374.
In summary, the number of salient pole pairs on the rotor of the synchronous machine is determined to be 374.
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(4) Why is belt drive usually arranged in high-speed class and chain drive generally at the low-speed in the design of the transmission system? (6scores) (5) What are the failure modes of gear transmission? (4 scores) (6) What are the design criteria for gear transmission? (5 scores)
(4) Belts possess high flexibility and elasticity which allow for smooth power transfer over long distances.
(5) Fatigue failures, wear failures, tooth fractures, and skipping teeth.
(6) Load capacity, material selection, transmission ratios, lubrication, and sound level.
Explanation:
In the design of a transmission system, the belt drive is usually arranged in the high-speed class while the chain drive is arranged in the low-speed class. This is because belts possess high flexibility and elasticity which allow for smooth power transfer over long distances.
Additionally, they have a low noise level, are long-lasting, and do not require frequent lubrication. Due to these features, belts are suitable for high-speed machinery.
On the other hand, chain drives are ideal for low-speed, high-torque applications. While they can transmit more power than belt drives, they tend to be noisier, less flexible, and require more lubrication. Hence, chain drives are best suited for low-speed applications.
The failure modes of gear transmission can be categorized into fatigue failures, wear failures, tooth fractures, and skipping teeth. Fatigue failures occur when a component experiences fluctuating loads, leading to cracking, bending, or fracture of the material. Wear failures happen when two parts rub against each other, resulting in material loss and decreased fit. Tooth fractures occur when high stress levels cause a tooth to break off. Skipping teeth, on the other hand, are caused by poor gear engagement, leading to the teeth skipping over one another, causing further wear and damage.
The design criteria for gear transmission include load capacity, material selection, transmission ratios, lubrication, and sound level. The load capacity refers to the ability to handle the transmitted load adequately. Material selection should consider factors such as sufficient strength, good machinability, good wear resistance, and corrosion resistance. The design must fulfill the transmission requirements such as speed and torque requirements. Lubrication is also critical as it helps reduce friction and wear. Finally, the noise level produced during gear transmission should be minimized.
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(10 pts) 9. A face milling operation removes 4.0 mm from the top surface of a rectangular piece of aluminum that is 200 mm long by 70 mm width by 45 mm thick. The cutter follows a path that is centered over the workpiece. It has four teeth and an 85-mm diameter. Cutting speed - 1.5 m/s, and chip load = 0.15 mm/tooth. Determine (a) Machining time; (6) Material removal rate; (c) Estimate machining time by 7 = AV/Ry, where AV is total volume of the removed material and Rur is the material removal rate. Is there any discrepancy between this result and the result in (a)? If so, what is the reason? Work Illustration of face milling in the cross-section view.
The given parameters are, Diameter of the cutter, D = 85mmChip load, h = 0.15mm/tooth Cutting speed, V = 1.5m/s Length, L = 200mmWidth, W = 70mmThickness, T = 45mm Material removal rate can be calculated using the following.
Where n is the rotational speed of the cutter. It can be calculated using the following formula, n = (1000 * V) / (π * D)n = (1000 × 1.5) / (π × 85)n = 55.527 rpm Now, putting all the values in the above formula, we get, Q = 0.15 * 4 * 85 * 55.527Q = 219.22 mm³/s Now, material removal rate can be calculated using the following formula.
A is the area of the cross-section of the workpiece. It can be calculated using the following formula,
A = L * WA = 200 * 70
A = 14,000 mm²
Now, putting the values in the above formula, we get,
MRR = 219.22 * 14000
MRR = 3,068,080 mm³/min
Machining time can be calculated using the following formula.
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Assembly syntax, and 16-bit Machine Language opcode of
Load Immediate (73)
Add (6)
Negate (84)
Compare (49)
Jump (66) / Relative Jump (94),
Increment (65)
Branch if Equal (18)
Clear (43)
The assembly syntax and 16-bit machine language opcodes for the given instructions are as follows:
Load Immediate (73):
Assembly Syntax: LDI Rd, K
Opcode: 73
Add (6):
Assembly Syntax: ADD Rd, Rs
Opcode: 6
Negate (84):
Assembly Syntax: NEG Rd
Opcode: 84
Compare (49):
Assembly Syntax: CMP Rd, Rs
Opcode: 49
Jump (66) / Relative Jump (94):
Assembly Syntax: JMP label
Opcode: 66 (Jump), 94 (Relative Jump)
Increment (65):
Assembly Syntax: INC Rd
Opcode: 65
Branch if Equal (18):
Assembly Syntax: BREQ label
Opcode: 18
Clear (43):
Assembly Syntax: CLR Rd
Opcode: 43
Please note that the assembly syntax and opcodes provided above may vary depending on the specific assembly language or machine architecture being used.
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A 3-phase industrial plant absorbs 52 kW at a line voltage of 480 V with a lagging power factor of 0.75 from a three-phase utility line. = 100 KVA and Vbase = 480 V, the per-unit voltage applied to the industrial plant is most nearly, If Sbase O a. 1 pu O b. 0.577 pu O c. 208 pu O d. 120 pu
Therefore, option A is correct.Option A: 1 pu, is the per-unit voltage applied to the industrial plant.
The solution is provided below;The apparent power, S is given by;
S = P / cosΦ... (i)
where P is the power in Watts and cosΦ is the power factor.
Now, the apparent power of the industrial plant is;
S = 52,000 / 0.75S
= 69,333.33 VA
= 69.333 kVA
The per-unit voltage applied to the industrial plant is most nearly given by;
pu = V / Vbase... (ii)
where V is the line voltage. Now, since the voltage is given as 480V, then;
pu = 480 / 480
= 1 pu
Therefore, option A is correct.Option A: 1 pu, is the per-unit voltage applied to the industrial plant.
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Design a circuit which counts seconds, minutes and hours and displays them on the 7-segement display in 24 hour format. The clock frequency available is 36 KHz. Assume that Binary to BCD converter and BCD to 7-Segement display is already available for the design.
The 24-hour clock has two digits for hours, two digits for minutes, and two digits for seconds. Binary Coded Decimal (BCD) is a technique for representing decimal numbers using four digits in which each decimal digit is represented by a 4-bit binary number.
A 7-segment display is used to display the digits from 0 to 9.
Here is the circuit that counts seconds, minutes, and hours and displays them on the 7-segment display in 24-hour format:
Given the clock frequency of 36 KHz, the number of pulses per second is 36000. The seconds counter requires 6 digits, or 24 bits, to count up to 59. The minutes counter requires 6 digits, or 24 bits, to count up to 59. The hours counter requires 5 digits, or 20 bits, to count up to 23.The clock signal is fed into a frequency divider that produces a 1 Hz signal. The 1 Hz signal is then fed into a seconds counter, minutes counter, and hours counter. The counters are reset to zero when they reach their maximum value.
When the seconds counter reaches 59, it generates a carry signal that increments the minutes counter. Similarly, when the minutes counter reaches 59, it generates a carry signal that increments the hours counter.
The outputs of the seconds, minutes, and hours counters are then converted to BCD format using a binary to BCD converter. Finally, the BCD digits are fed into a BCD to 7-segment display decoder to produce the display on the 7-segment display.Here's a block diagram of the circuit: Block diagram of the circuit
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A 2 DOF system has mode shapes given by Φ₁ = {1}
{-2}
and
Φ₂ =
{1}
{3}
A force vector F = {1}
{p}
sin(Ωt) is acting on the system. Find the value of P if the system steady state response is purely in mode 1.
A 2 DOF (Degree of Freedom) system has mode shapes given by Φ₁ = {1} {-2} and Φ₂ = {1} {3}. A force vector F = {1} {p} sin(Ωt) is acting on the system, where P is the value of the steady-state response in mode
1.The system response can be given by the equation,
M = M₀ + M₁ sin(Ωt + φ₁) + M₂ sin(2Ωt + φ₂)
Here,Ω = 1 (the driving frequency)
φ₁ is the phase angle of the first modeφ₂ is the phase angle of the second modeM₀ is the static deflection
M₁ is the amplitude of the first mode
M₂ is the amplitude of the second mode
So, the response of the system can be given by:
M = M₁ sin(Ωt + φ₁)
Now, substituting the values,
M = Φ₁ F = {1} {-2} {1} {p} sin(Ωt) = {1-2p sin(Ωt)}
In order for the steady-state response to be purely in mode 1, M₂ = 0
So, the equation for the response becomes,
M = M₁ sin(Ωt + φ₁) ⇒ {1-2p sin(Ωt)} = M₁ sin(Ωt + φ₁)
Comparing both sides, we get,
M₁ sin(Ωt + φ₁) = 1 and -2p sin(Ωt) = 0sin(Ωt) ≠ 0, as Ω = 1, so -2p = 0P = 0
Therefore, the value of P if the system steady-state response is purely in mode 1 is 0
In this problem, we are given a 2 DOF (Degree of Freedom) system having mode shapes Φ₁ and Φ₂.
The mode shapes of a system are the deflected shapes that result from the system vibrating in free vibration. In the absence of any external forcing, these deflected shapes are called natural modes or eigenmodes. The system is also subjected to a force vector F = {1} {p} sin(Ωt).
We have to find the value of P such that the system's steady-state response is purely in mode 1. Steady-state response refers to the long-term behavior of the system after all the transient vibrations have decayed. The steady-state response is important as it helps us predict the system's behavior over an extended period and gives us information about the system's durability and reliability.
In order to find the value of P, we first find the system's response. The response of the system can be given by the equation,
M = M₀ + M₁ sin(Ωt + φ₁) + M₂ sin(2Ωt + φ₂)
where M₀, M₁, and M₂ are constants, and φ₁ and φ₂ are the phase angles of the two modes.
In this case, we are given that Ω = 1 (the driving frequency), and we assume that the system is underdamped. Since we want the steady-state response to be purely in mode 1, we set M₂ = 0.
Hence, the equation for the response becomes,
M = M₁ sin(Ωt + φ₁)
We substitute the values of Φ₁ and F in the above equation to get,{1-2p sin(Ωt)} = M₁ sin(Ωt + φ₁)
Comparing both sides, we get,
M₁ sin(Ωt + φ₁) = 1 and -2p sin(Ωt) = 0sin(Ωt) ≠ 0, as Ω = 1, so -2p = 0P = 0
Therefore, the value of P if the system steady-state response is purely in mode 1 is 0.
The value of P such that the system steady-state response is purely in mode 1 is 0.
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In a reheat cycle steam at 15 MPa, 540°C enters the engine and expands to 1.95 MPa. At this point the steam is withdrawn and passed through a reheater. It reenters the engine at 540°C. Expansion now occurs to the condenser pressure of 0.0035 MPa.
(a) For the ideal cycle, find ee.
(b) A 60,000 kw turbine operates between the same state points except that the steam enters the reheater at 1.95 MPa and 260°C, departs at 1.8 MPa and 540°C. The steam flow is 147,000 kg/hr, generator efficiency is 96%. For actual engine, find, ek, mk, and nk,
(c) Determine the approximate enthalpy of the exhaust steam if the heat lost through the turbine casing is 2% of the combined work.
a) Therefore, ideal efficiency is 61.3% and b) 96% actual engine and c) The approximate enthalpy of the exhaust steam if the heat lost through the turbine casing is 2% of the combined work is H4 = 171.9 kJ/kg.
a. For the ideal cycle, the efficiency can be calculated as follows;
Efficiency,η = (1 - T2/T1)where T2 is the temperature at the exhaust and T1 is the temperature at the inlet of the engine.
The state points can be read off the Mollie diagram for steam.
The state points are;
State 1: Pressure = 15 MPa, Temperature = 540°C
State 2: Pressure = 1.95 MPa, Temperature = 316°C
State 3: Pressure = 0.0035 MPa, Temperature = 41.6°CT1 = 540 + 273 = 813 K, T2 = 41.6 + 273 = 314.6 Kη = (1 - 314.6/813)η = 61.3%
Therefore, ideal efficiency is 61.3%.
b. For an actual engine;
Generator output = 60,000 kW = Work done/second = m × (h1 - h2)
where m is the steam flow rate in kg/hr, h1 and h2 are the specific enthalpies at state 1 and state 2.
The steam flow is given as 147,000 kg/hr.h1 = 3279.3 kJ/kg, h2 = 2795.4 kJ/kg
Power supplied to the turbine= 60,000/0.96= 62,500 kW = Work done/second = m × (h1 - h2a)where h2a is the specific enthalpy at state 2a and m is the steam flow rate in kg/hr.
The specific enthalpies at state 2a can be found from the Mollier diagram, as follows;
At 1.95 MPa and 260°C, h2s = 2865.7 kJ/kg
At 1.8 MPa and 540°C, h2a = 3442.9 kJ/kg
Power loss in the engine, wk = 62500 - 60000 = 2500 kW
Also, m = 147,000/3600= 40.83 kg/s
Work output of the engine = m × (h1 - h3)where h3 is the specific enthalpy at state 3. h3 can be read from the Mollier diagram as 194.97 kJ/kg.
Total work done = Work output + Work loss = m × (h1 - h3) + wk
The efficiency of the engine can be calculated as follows;η = (Work output + Work loss)/Heat supplied
Heat supplied = m × (h1 - h2s)η = ((m × (h1 - h3)) + wk)/(m × (h1 - h2s))
The mass flow rate m is 40.83 kg/s;
h1 = 3279.3 kJ/kg, h2s = 2865.7 kJ/kg, h3 = 194.97 kJ/kgw
k = 2500 kWη = ((40.83 × (3279.3 - 194.97)) + 2500)/((40.83 × (3279.3 - 2865.7))η = 36.67%
For an actual engine;
ek = 36.67%mk = 40.83 kg/snₖ = 96%
In a Reheat cycle, the enthalpy of the exhaust steam if the heat lost through the turbine casing is 2% of the combined work can be calculated as follows:
Heat rejected from the turbine casing = 2% of the combined work done= 2/100 * (m(h1 - h3) + wk)
The enthalpy of the exhaust steam is calculated as follows;
H4 = h3 - (Heat rejected from the turbine casing/m)
H4 = 194.97 - (0.02(m(h1 - h3) + wk)/m)
H4 = 171.9 kJ/kg
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Slider crank kinematic and force analysis. Plot of input and
output angles.
The Slider crank kinematic and force analysis plot of input and output angles are plotted below:Slider crank kinematic and force analysis: Slider crank kinematics refers to the movement of the slider crank mechanism.
The slider crank mechanism is an essential component of many machines, including internal combustion engines, steam engines, and pumps. Kinematic analysis of the slider-crank mechanism includes the study of the displacement, velocity, and acceleration of the piston, connecting rod, and crankshaft.
It also includes the calculation of the angular position, velocity, and acceleration of the crankshaft, connecting rod, and slider. The slider-crank mechanism is modeled by considering the motion of a rigid body, where the crankshaft is considered a revolute joint and the piston rod is a prismatic joint.
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If the production of a material increases by r% every year, show
that the doubling time is given by 70/r.
When the production of a material increases at the rate of r% every year, the doubling time is given by 70/r. Assume that the initial production rate is P₀ at the start of the year, and after t years, it will be P.
After the first year, the production rate will be
P₁ = P₀ + (r/100)P₀
P₁ = (1 + r/100)P₀.
In general, the production rate after t years is given by the formula
P = (1 + r/100)ᵗP₀.
when the production of a material is doubled, the following equation is satisfied:
2P₀ = (1 + r/100)ᵗP₀
Applying the logarithm to both sides of the equation, we obtain:
log 2 = tlog(1 + r/100)
Dividing both sides by log(1 + r/100), we get:
t = log 2 / log(1 + r/100)
This expression shows the number of years required for the production of a material to double at a constant rate of r% per year. Using the logarithm property, we can rewrite the above equation as:
t = 70/ln(1 + r/100)
In the above expression, ln is the natural logarithm.
By substituting ln(2) = 0.693 into the equation, we can obtain:
t = 0.693 / ln(1 + r/100)
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True/fase
4. Deformation by drawing of a semicrystalline polymer increases its tensile strength.
5.Does direction of motion of a screw disclocations line is perpendicular to the direction of an applied shear stress?
6.How cold-working effects on 0.2% offself yield strength?
4. False. Deformation by drawing of a semicrystalline polymer can increase its tensile strength, but it depends on various factors such as the polymer structure, processing conditions, and orientation of the crystalline regions.
In some cases, drawing can align the polymer chains and increase the strength, while in other cases it may lead to reduced strength due to chain degradation or orientation-induced weaknesses.
5. True. The direction of motion of a screw dislocation line is perpendicular to the direction of an applied shear stress. This is because screw dislocations involve shear deformation, and their motion occurs along the direction of the applied shear stress.
6. Cold working generally increases the 0.2% offset yield strength of a material. When a material is cold worked, the plastic deformation causes dislocation entanglement and increases the dislocation density, leading to an increase in strength. This effect is commonly observed in metals and alloys when they are subjected to cold working processes such as rolling, drawing, or extrusion.
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At a post office, customers wait in a single line for the first open window. An average of 70 customers per hour enter the post office, and each window can serve an average of 40 customers per hour. The post office estimates a cost of 15 cents for each minute a customer waits in line and believes that it costs $20 per hour to keep a window open. Interarrival times and service times are exponential. To minimize the total expected hourly cost, how many windows should be open?
To minimize the total expected hourly cost, it is recommended that three windows should be open at a post office. The customers wait in a single line for the first open window.
Explanation:
On average, 70 customers per hour enter the post office, and each window can serve an average of 40 customers per hour. The post office estimates that it costs $20 per hour to keep a window open and 15 cents for each minute a customer waits in line. Interarrival times and service times are exponential.
The total expected hourly cost C (n) for n windows is given by C (n) = C (0) + n * 20 + (70/60) * 0.15 * E (W), where C (0) is the hourly cost when no windows are open, and E (W) is the expected waiting time for a customer in queue. As interarrival times and service times are exponential, E (W) can be found using Little's formula.
E (W) = E (N) / (70/60), where E (N) is the expected number of customers in the queue. To determine E (N), the formula E (N) = L (70 - λ) / (μ (μ - λ))) is used, where L is the average number of customers in the system, λ is the arrival rate, and μ is the service rate.
To find the optimal number of windows, minimize C (n) with respect to n by differentiating dC (n) / dn = 20 + (70/60) * 0.15 * (dE (N) / dn) = 0. Simplifying the equation gives dE (N) / dn = - (240/7) * n + (210/7). Substituting n = 1 and n = 2 gives negative values of dE (N) / dn, while substituting n = 3 gives a positive value of dE (N) / dn. Therefore, the optimal number of windows is three (3).
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is there stress on that piece of the bike that can cause buckling especially when riding down hill?
Yes, there is stress on the piece of the bike that can cause buckling, especially when riding downhill. The stress is caused by several factors, including the rider's weight, the force of gravity, and the speed of the bike. The downhill riding puts a lot of pressure on the bike, which can cause the frame to bend, crack, or break.
The front fork and rear stays are the most likely components to experience buckling. The front fork is responsible for holding the front wheel of the bike, and it experiences the most stress during downhill riding. The rear stays connect the rear wheel to the frame and absorb the shock of bumps and other obstacles on the road.
To prevent buckling, it is essential to ensure that your bike is in good condition before heading downhill. Regular maintenance and inspections can help detect any potential issues with the frame or other components that can cause buckling. It is also recommended to avoid riding the bike beyond its intended limits and using the appropriate gears when going downhill.
Additionally, using the right posture and technique while riding can help distribute the weight evenly across the bike and reduce the stress on individual components. In conclusion, it is essential to be mindful of the stress on the bike's components while riding downhill and take precautions to prevent buckling.
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The temperature in a shower is adjusted by the ratio of 'cold' to 'hot' water. The temperature of the hot water supply is 140 F and the desired water temperature of the outflow from the shower head is 100 F. Determine the required ratio m_c/m_h (mass flow rate of cold water/mass flow rate of hot water) if the cold water supply is a) 40F and b) 80F.
Assume steady state steady flow, m_total is a constant, and ICL.
This ratio adjusts the temperature in a shower by the proportion of cold water to hot water.
Hence, we have:
m_total = m_h + m_c
Q_h = m_h * h_fg
Q_c = m_c * h_fg
The heat transfer rate from the hot water to the cold water can be calculated as:
Q_h = m_h * c * (h_o - h_i)
where c is the specific heat of water and h_i and h_o are the enthalpies of the hot water at the inlet and outlet, respectively.
Given T_c = 80°F, we can calculate the ratio m_c/m_h (mass flow rate of cold water/mass flow rate of hot water) for cold water supplies at 40°F and 80°F.
For T_c = 40°F:
m_c/m_h = (140 - 100)/(100 - 40) = 2.5
For T_c = 80°F:
m_c/m_h = (140 - 100)/(100 - 80) = 2.5
Therefore, the required ratio m_c/m_h for cold water supplies at 40°F and 80°F is 2.5.
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An industrial plant absorbs 500 kW at a line voltage of 480 V with a lagging power factor of 0.8 from a three-phase utility line. The current absorbed from the utility company is most nearly O a. 601.4 A O b. 281.24 A O c. 1041.67 A O d. 751.76 A
The current absorbed from the utility company is most nearly 601.4 A (Option A).Hence, the correct option is (A) 601.4 A.
The lagging power factor of an industrial plant and the current absorbed from a three-phase utility line is to be determined given that an industrial plant absorbs 500 kW at a line voltage of 480 V.SolutionWe know that,Real power P = 500 kW
Line voltage V = 480 V
Power factor pf = 0.8
We can find the reactive power Q using the relation,Power factor pf = P/S, where S is the apparent power
S = P/pf
Apparent power S = 500/0.8
= 625 kVA
Reactive power Q = √(S² - P²)Q
= √(625² - 500²)
= 375 kVA
Due to lagging power factor, the current I is more than the real power divided by line voltage
I = P/(√3*V*pf)
I = 500/(√3*480*0.8)
I = 601.4 A
Now, the current absorbed from the utility company is most nearly 601.4 A (Option A).Hence, the correct option is (A) 601.4 A.
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Question A pendulum has a length of 250mm. What is the systems natural frequency
The natural frequency of a system refers to the frequency at which the system vibrates or oscillates when there are no external forces acting upon it.
The natural frequency of a pendulum is dependent upon its length. Therefore, in this scenario, a pendulum has a length of 250 mm and we want to find its natural frequency.Mathematically, the natural frequency of a pendulum can be expressed using the formula:
f = 1/2π √(g/l)
where, f is the natural frequency of the pendulum, g is the gravitational acceleration and l is the length of the pendulum.
Substituting the given values into the formula, we get :
f= 1/2π √(g/l)
= 1/2π √(9.8/0.25)
= 2.51 Hz
Therefore, the natural frequency of the pendulum is 2.51 Hz. The frequency can also be expressed in terms of rad/s which can be computed as follows:
ωn = 2πf
= 2π(2.51)
= 15.80 rad/s.
Hence, the system's natural frequency is 2.51 Hz or 15.80 rad/s. This is because the frequency of the pendulum is dependent upon its length and the gravitational acceleration acting upon it.
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A. Considering a stoichiometric mixture of the fuel given below (C 3H 1O 2) and air, determine a. The stoichiometric reaction equation. b. The mole fraction of oxygen (O2). c. The air-fuel ratio.
The stoichiometric reaction equation for the fuel C3H10 and air is C3H10 + (13/2)O2 -> 3CO2 + 5H2O. The mole fraction of oxygen (O2) can be calculated by dividing the moles of O2 by the total moles of the mixture.
The air-fuel ratio is determined by dividing the moles of air (oxygen) by the moles of fuel, and in this case, it is 6.5:1.
a. The stoichiometric reaction equation for the fuel C3H10 is:
C3H10 + (13/2)O2 -> 3CO2 + 5H2O
b. To determine the mole fraction of oxygen (O2), we need to calculate the moles of oxygen relative to the total moles of the mixture. In the stoichiometric reaction equation, the coefficient of O2 is (13/2). Since the stoichiometric ratio is based on the balanced equation, the mole fraction of O2 can be calculated by dividing the moles of O2 by the total moles of the mixture.
c. The air-fuel ratio can be calculated by dividing the moles of air (oxygen) by the moles of fuel. In this case, the stoichiometric reaction equation indicates that 13/2 moles of O2 are required for 1 mole of C3H10. Therefore, the air-fuel ratio can be expressed as 13/2:1 or 6.5:1.
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