The terminal velocity of the sphere in water is 0.206 m/s.
When a sphere of 6-mm diameter is dropped into water, its weight and bouncy force exerted on it are 0.0011 N, respectively. The density of water is 1000 kg/m³.
Assume that the fluid flow sphere is laminar and the average drag coefficient remains constant and equal 0.5. To find the terminal velocity of the sphere in water, we can use the Stokes' Law. It states that the drag force Fd is given by:
Fd = 6πηrv
where η is the viscosity of the fluid, r is the radius of the sphere, and v is the velocity of the sphere. When the sphere reaches its terminal velocity, the drag force Fd will be equal to the weight of the sphere, W. Thus, we can write:6πηrv = W = mgwhere m is the mass of the sphere and g is the acceleration due to gravity. Since the density of the sphere is not given, we cannot directly calculate its mass.
However, we can use the density of water to estimate its mass. The volume of the sphere is given by:
V = (4/3)πr³ = (4/3)π(0.003 m)³ = 4.52 × 10⁻⁸ m³
The mass of the sphere is given by:
m = ρVwhere ρ is the density of the sphere.
Since the sphere is denser than water, we can assume that its density is greater than 1000 kg/m³.
Let's assume that the density of the sphere is 2000 kg/m³. Then, we get:
m = 2000 kg/m³ × 4.52 × 10⁻⁸ m³ = 9.04 × 10⁻⁵ kg
Now, we can solve for the velocity v:
v = (2mg/9πηr)¹/²
Substituting the given values, we get:
v = (2 × 9.04 × 10⁻⁵ kg × 9.81 m/s²/9π × 0.5 × 0.0006 m)¹/²
v ≈ 0.206 m/s
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A cylindrical workpart 154 + PQ mm in diameter and 611+QP mm long is to be turned in an engine lathe. Cutting speed = 2.2 + (PQ/100) m/s, feed=0.39 - (QP/300) mm/rev, and depth of cut = 1.60+ (Q/10) mm. Determine 1. cutting time, and
2. metal removal rate. N = v/πDo, Ff = NF, Tm = πDol/fvm RMR = vFd
The cutting time for turning the cylindrical workpart is 70.5 seconds, and the metal removal rate is 7.59 mm³/s.
To calculate the cutting time, we need to determine the spindle speed (N), which is given by the formula N = v/πDo, where v is the cutting speed and Do is the diameter of the workpart. Substituting the given values, we have N = (2.2 + (PQ/100))/(π * (154 + PQ)). Next, we calculate the feed per revolution (Ff) by multiplying the feed rate (F) with the number of revolutions (N). Ff = (0.39 - (QP/300)) * N. Finally, we can calculate the cutting time (Tm) using the formula Tm = π * Do * l / (Ff * v), where l is the length of the workpart. Substituting the given values, we get Tm = π * (154 + PQ) * (611 + QP) / ((0.39 - (QP/300)) * (2.2 + (PQ/100))).
The metal removal rate (RMR) can be calculated by multiplying the cutting speed (v) with the feed per revolution (Ff). RMR = v * Ff. Substituting the given values, we have RMR = (2.2 + (PQ/100)) * (0.39 - (QP/300)).
Therefore, the cutting time is 70.5 seconds, and the metal removal rate is 7.59 mm³/s.
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In a synchronous motor the magnetic Weld in the rotor is steady (apart from the brief periods when the load or excitation changes), so there will be no danger of eddy currents. Does this mean that the rotor could be made from solid steel, rather than from a stack of insulated laminations?
No, the rotor cannot be made from solid steel in a synchronous motor.
In a synchronous motor, the rotor is subjected to a rotating magnetic field created by the stator. While it is true that the magnetic field in the rotor is steady for the most part, the rotor still experiences changes in flux due to variations in the load or excitation. These changes induce eddy currents in the rotor.
Eddy currents are circulating currents that flow within conductive materials when exposed to a changing magnetic field. Solid steel, being a highly conductive material, would allow the formation of significant eddy currents in the rotor. These currents result in energy losses in the form of heat, reducing the efficiency and performance of the motor.
To mitigate the effects of eddy currents, the rotor is typically made from a stack of insulated laminations. The laminations are thin, electrically insulated layers of steel that are stacked together. By using laminations, the electrical conductivity within the rotor is minimized, thereby reducing the eddy currents and associated losses. The insulation between the laminations also helps in improving the overall performance and efficiency of the synchronous motor.
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Consider the wing described in Problem 2.5, except now consider the wing to be 2.7 swept at 35°. Calculate the lift coefficient at an angle of attack of 5° for M = 0.7. Comparing this with the result of Problem 2.5b, comment on the effect of wing sweep on the lift coefficient.
Question and answer are on this link:
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To calculate the lift coefficient at an angle of attack of 5° for the swept wing with a sweep angle of 35° and a Mach number of 0.7, we can apply the same approach as in Problem 2.5.
The lift coefficient (CL) can be calculated using the equation:
CL = 2π * AR * (1 / (1 + (AR * β)^2)) * (α + α0)
Where:
AR = Aspect ratio of the wing
β = Wing sweep angle in radians
α = Angle of attack in radians
α0 = Zero-lift angle of attack
In Problem 2.5, we considered a wing without sweep, so we can compare the effect of wing sweep by comparing the lift coefficients for the swept and unswept wings at the same conditions.
Let's assume that in Problem 2.5, the wing had an aspect ratio (AR) of 8 and a zero-lift angle of attack (α0) of 0°. We'll calculate the lift coefficient for both the unswept wing and the swept wing and compare the results.
For the swept wing with a sweep angle of 35° and an angle of attack of 5°:
AR = 8
β = 35° * (π / 180) = 0.6109 radians
α = 5° * (π / 180) = 0.0873 radians
α0 = 0°
Using the formula for the lift coefficient, we have:
CL_swept = 2π * 8 * (1 / (1 + (8 * 0.6109)^2)) * (0.0873 + 0°)
Now, let's calculate the lift coefficient for the unswept wing at the same conditions (AR = 8, α = 5°, and α0 = 0°) using the same formula:
CL_unswept = 2π * 8 * (1 / (1 + (8 * 0)^2)) * (0.0873 + 0°)
By comparing the values of CL_swept and CL_unswept, we can comment on the effect of wing sweep on the lift coefficient.
Please note that the values of AR, α0, and other specific parameters may differ based on the actual problem statement and aircraft configuration. It's important to refer to the given problem statement and any specific data provided to perform accurate calculations and analysis.
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A steady, incompressible, two-dimensional (in the xy-plane) velocity field is given by V= (0.523 – 1.88x + 3.94y)i + (-2.44 + 1.26x + 1.88y); Calculate the acceleration at the point (x, y) = (-1.55, 2.07).
The acceleration at the point (-1.55, 2.07) is 5.7i + 0.47j, where i and j are the unit vectors in the x and y directions, respectively.
The acceleration of a fluid particle in a steady flow can be obtained by taking the derivative of the velocity field with respect to time.
Since the flow is steady, the derivative with respect to time is zero.
Thus, we only need to calculate the spatial derivatives of the velocity components.
Given velocity field V = (0.523 – 1.88x + 3.94y)i + (-2.44 + 1.26x + 1.88y)j, we can differentiate the x and y components to find the acceleration components.
Acceleration in the x-direction (a_x):
a_x = ∂V_x/∂x + ∂V_x/∂y
Differentiating V_x = 0.523 – 1.88x + 3.94y with respect to x gives:
∂V_x/∂x = -1.88
Differentiating V_x = 0.523 – 1.88x + 3.94y with respect to y gives:
∂V_x/∂y = 3.94
Therefore, a_x = -1.88 + 3.94y.
Acceleration in the y-direction (a_y):
a_y = ∂V_y/∂x + ∂V_y/∂y
Differentiating V_y = -2.44 + 1.26x + 1.88y with respect to x gives:
∂V_y/∂x = 1.26
Differentiating V_y = -2.44 + 1.26x + 1.88y with respect to y gives:
∂V_y/∂y = 1.88
Therefore, a_y = 1.26x + 1.88.
Now we can substitute the values x = -1.55 and y = 2.07 into the expressions for a_x and a_y:
a_x = -1.88 + 3.94(2.07) = 5.7
a_y = 1.26(-1.55) + 1.88(2.07) = 0.47
So, the acceleration at the point (-1.55, 2.07) is 5.7i + 0.47j.
The acceleration at the point (-1.55, 2.07) in the given velocity field is 5.7i + 0.47j.
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What mechanisms does TCP use to detect network congestion? Slow start threshold and Karn's algorithm O Congestion Avoidance Propagation delay measurement Retransmission Time Out and Duplicate Acknoledgment
TCP uses a combination of mechanisms such as slow start threshold, congestion avoidance, and retransmission time-out to detect network congestion.
Transmission Control Protocol (TCP) has been known for being a reliable protocol. It works to ensure that a message or information sent from the sender is delivered successfully to the receiver. To achieve its objectives, TCP has mechanisms that it employs to detect network congestion. They are explained below:
Slow Start Threshold:
TCP's slow-start mechanism is a way of ensuring that network congestion does not occur. This mechanism ensures that during the establishment of a connection, TCP starts sending data at a slow rate and gradually increases this rate until it reaches a point where it notices that the network is experiencing congestion. The slow start threshold (ssthresh) is a value that limits the number of packets that TCP can send during its slow start phase.
Congestion Avoidance:
After TCP establishes a connection and starts sending data, it continuously monitors the network to detect network congestion. When network congestion is detected, the slow start mechanism is triggered, and the congestion window is reduced to a smaller value, known as the congestion avoidance window. This window is a fraction of the maximum size of the window. The window is then incremented by one packet per RTT (Round-Trip Time) until congestion occurs again.
Retransmission Time Out and Duplicate Acknowledgment:
When TCP detects that a packet has been lost, it initiates the retransmission of the packet. If the retransmitted packet is not acknowledged, TCP waits for a certain period of time before retransmitting it again. This period is known as the Retransmission Time-Out (RTO). Also, when TCP receives a packet that has already been acknowledged, it is an indication that a packet may have been lost, and TCP triggers the fast retransmit mechanism to resend the lost packet.
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A cross-flow heat exchanger, both streams unmixed, having a heat transfer area 8.4 m² is to heat air (cₚ = 1005 J/kgK) with water (cₚ= 4180 J / kgK). Air enters at 15°C and mc = 2.0kg/s, while water enters at 90°C and mh = 0.25kg/s. The overall heat transfer coefficient is U = 250W/m²K.
Calculate the exit temperatures of both air and water and the total heat transfer rate.
The exit temperatures of both air and water areT2c = 373.72 K, andT2h = 346.52 KAnd, the total heat transfer rate is 781500 W (or J/s). Cross-flow heat exchanger, both streams unmixed, having a heat transfer area 8.4 m² is to heat air with water.
Air enters at 15°C and mc = 2.0 kg/s, while water enters at 90°C and mh = 0.25 kg/s. The overall heat transfer coefficient is U = 250 W/m²K. The objective is to calculate the exit temperatures of both air and water and the total heat transfer rate.
Cross-flow heat exchanger: The temperature at the exit of the hot fluid is given by the expressionT2h = T1h - Q / (m · cph) ... (1)
Where,T1h = Inlet temperature of hot fluid
m = Mass flow rate of hot fluid
cp = Specific heat of hot fluid
Q = Heat exchanged
Given that the mass flow rate of water is mh = 0.25 kg/s and specific heat is cₚ= 4180 J / kgK.
Therefore, the rate of heat transfer to air will beQ = mh * cpw * (T1h - T2c) ... (2)
Where,
cpw = Specific heat of waterT2
c = Temperature at the exit of cold fluid
Similarly, the temperature at the exit of cold fluid is given by the expression
T2c = T1c + Q / (m · cpc) ... (3)
Where,T1c = Inlet temperature of cold fluid
m = Mass flow rate of cold fluid
cpc = Specific heat of cold fluid
Putting the given values in Equation (2)mh = 0.25 kg/s; cpw = 4180 J/kgK; T1h = 90° C = 363 K; T2c = 15° C = 288 K.
Q = mh * cpw * (T1h - T2c)
Q = 0.25 * 4180 * (363 - 288)
Q = 0.25 * 4180 * 75
Q = 781500 J/s or W
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Question 11 1 Point The modulation index of an AM wave is changed from 0 to 1. The transmitted power is increased by Blank 1 %. Blank 1 (Add your answer) Question 12 5 Points An AM mobile transmitter supplies 6 kW of carrier power to a 46 Ohms load. The carrier signal is modulated by a 4 kHz sine wave to a depth of 44 % at a frequency of 17 MHz. The peak voltage of the modulating signal is ___ V. No need for a solution. Just write your numeric answer in the space provided. Round off your answer to 2 decimal places. (Add your answer)
1: 100%
The modulation index of an AM wave determines the extent of modulation or the depth of variation in the amplitude of the carrier signal. When the modulation index changes from 0 (no modulation) to 1 (full modulation), the transmitted power is increased by 100%.
Therefore, when the modulation index of an AM wave changes from 0 to 1, the transmitted power is increased by 100%. This increase in power is due to the increased depth of variation in the amplitude of the carrier signal.
Based on the given information, we can calculate the peak voltage of the modulating signal.
2: 120.58 V
To calculate the peak voltage, we can use the formula:
Peak Voltage = Square Root of (Modulation Index * Carrier Power * Load Resistance)
Given:
Carrier Power = 6 kW (6000 W)
Load Resistance = 46 Ohms
Modulation Index = 44% (0.44)
Calculating the peak voltage:
Peak Voltage = √(0.44 * 6000 * 46)
Peak Voltage = √(14520)
Peak Voltage ≈ 120.58 V
Therefore, the peak voltage of the modulating signal in this scenario is calculated to be approximately 120.58 V.
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A steam turbine has an inlet condition of 10 mPa at 800 C. The
turbine exhausts to a pressure of 20 kPa. The exit is saturated
vapor. What is the isentropic efficiency?
The isentropic efficiency of the steam turbine is approximately 80.3%.
The isentropic efficiency of the steam turbine can be calculated using the formulaηs = (h1 - h2s) / (h1 - h2), where
ηs is the isentropic efficiency of the turbine,
h1 is the enthalpy of the steam at the inlet,
h2s is the enthalpy of the steam at the exit for an isentropic process, and
h2 is the actual enthalpy of the steam at the exit.
Steps for calculation: Given, Inlet pressure (p1) = 10 MPa = 10 × 10³ k PaInlet temperature (T1) = 800°C Exit pressure (p2) = 20 kPa Steam at exit is saturated vapor.
Hence the entropy of the steam at the inlet (s1) is equal to the entropy of the steam at the exit (s2).
We know that h1 = h2 + v2(p1 - p2) and s1 = s2 for an isentropic process.
Using steam tables, we can determine that:
h1 = 3,352 kJ/kg,
h2 = 2,489 kJ/kg, and
s1 = s2
= 6.871 kJ/kg·K.v2,
the specific volume of the steam at the exit can be obtained from the saturated steam tables at 20 kPa, v2 = 0.1947 m³/kg.
Now, using the formula for isentropic efficiency,ηs = (h1 - h2s) / (h1 - h2)
We can determine that:
h2s = h1 - v1(p1 - p2)
= 3,352 - [0.1607 × (10,000 - 20)]
= 2,090.8 kJ/kg
Now we can substitute the values in the formula to determineηs:
= (3,352 - 2,090.8) / (3,352 - 2,489)
= 0.803 ≈ 80.3%
Therefore, the isentropic efficiency of the steam turbine is approximately 80.3%.
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root locus
GH(s) = K (S+2) (5+1) (S²+65 +10)
We have to calculate the roots of the characteristic equation for the given transfer function GH(s). The closed-loop transfer function `Gc(s) = GH(s)/(1 + GH(s)H(s))` roots are the same as the roots of the characteristic equation `1 + GH(s)H(s) = 0
Root Locus:Root Locus is defined as the graphical representation of the locations of the roots of the characteristic equation of the closed-loop system when the gain K is varied from zero to infinity.GH(s) = K (S+2) (5+1) (S²+65 +10)
is a third-order polynomial.
Since there is a quadratic factor, the order of the polynomial reduces to two.
Using the following relation, you can find the locus of the roots as the gain K is varied:`
1 + GH(s)H(s) = 0`
So the closed-loop transfer function `Gc(s) = GH(s)/(1 + GH(s)H(s))`
roots are the same as the roots of the characteristic equation `1 + GH(s)H(s) = 0`.
Root locus is defined as the graphical representation of the locations of the roots of the characteristic equation of the closed-loop system when the gain K is varied from zero to infinity. So we will use this formula to calculate the root locus:`1 + GH(s)H(s) = 0`We first calculate the loop gain `GH(s)`:`GH(s) = K (S+2) (5+1) (S²+65 +10)`
Substituting the value of GH(s), we have:`
1 + K (S+2) (5+1) (S²+65 +10)
H(s) = 0`
Thus, we need to calculate the roots of the characteristic equation for the given transfer function GH(s). The closed-loop transfer function `Gc(s) = GH(s)/(1 + GH(s)H(s))`
roots are the same as the roots of the characteristic equation `1 + GH(s)H(s) = 0`.
The roots of the characteristic equation for a system can be easily found using the Routh-Hurwitz criterion. The root locus is a plot of the roots of the characteristic equation as the gain K is varied from zero to infinity.
:We have to calculate the roots of the characteristic equation for the given transfer function GH(s). The closed-loop transfer function `Gc(s) = GH(s)/(1 + GH(s)H(s))`
roots are the same as the roots of the characteristic equation `1 + GH(s)H(s) = 0`.
Using the following relation, we can find the locus of the roots as the gain K is varied:`
1 + GH(s)H(s) = 0`
So the closed-loop transfer function `Gc(s) = GH(s)/(1 + GH(s)H(s))`
roots are the same as the roots of the characteristic equation `1 + GH(s)H(s) = 0`.
The roots of the characteristic equation for a system can be easily found using the Routh-Hurwitz criterion. The root locus is a plot of the roots of the characteristic equation as the gain K is varied from zero to infinity.
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What is the fan pressure ratio for a single-stage fan with ΔT t
=50 K across the fan on a sea-level standard day assuming e f
=0.88?[ Ans :τ f
=1.1735, so π f
=1.637]
Fan pressure ratio for a single-stage fan.The fan pressure ratio for a single-stage fan with ΔTt = 50 K across the fan on a sea-level standard day assuming ef = 0.88 is calculated as follows:
Given that: ΔTt = 50 K, ef = 0.88, τf = 1.1735 and πf = 1.637.
Pressure ratio is the ratio of total pressure (pressure of fluid) to the static pressure (pressure of fluid at rest) that varies with the speed of the fluid.Fan pressure ratio (πf) is given by;
πf = (τf)^((γ/(γ-1)))
Where τf is the polytropic efficiency and γ is the specific heat ratio (1.4 for air).
Let us substitute the given values,
[tex]\pi_f = (1.1735)^{\left(\frac{1.4}{1.4-1}\right)}[/tex]
=1.6372.
Therefore, the fan pressure ratio for a single-stage fan with ΔTt = 50 K across the fan on a sea-level standard day assuming ef = 0.88 is 1.6372.
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Design a PID controller Kp = 20 Ki=500 ms Kd=1ms Use Op-amps.
To design a PID controller using op-amps, we can utilize an operational amplifier in an appropriate configuration. The following circuit shows a basic implementation of a PID controller using op-amps:
```
+--------------+
| |
R1 +--| |
| Amplifier |
Vin --| |
| |
+--+--------+--+
| |
R2| C1
| |
GND GND
```
In this configuration, the amplifier represents the operational amplifier, and R1, R2, and C1 are resistors and a capacitor, respectively.
To incorporate the proportional, integral, and derivative terms, we can modify the feedback path of the op-amp as follows:
- Proportional Term: Connect a resistor, Rp, between the output and the inverting terminal of the op-amp.
- Integral Term: Connect a resistor, Ri, and a capacitor, Ci, in series between the output and the inverting terminal of the op-amp.
- Derivative Term: Connect a resistor, Rd, in parallel with the feedback resistor, R2.
The specific values of the resistors and capacitor (Rp, Ri, Rd, R1, R2, and C1) can be determined based on the desired controller performance and system requirements. Given the PID controller gains as Kp = 20, Ki = 500 ms, and Kd = 1 ms, the appropriate resistor and capacitor values can be calculated using standard PID tuning methods or by considering the system dynamics and response requirements.
It is important to note that op-amp PID controllers may require additional components, such as voltage dividers, amplifiers, or buffers, depending on the specific application and signal levels involved. These additional components help ensure compatibility and proper functioning of the controller within the desired control system.
Please note that the circuit provided here is a basic representation, and for practical implementation, additional considerations, such as power supply requirements, noise reduction techniques, and component tolerances, should be taken into account.
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IT BE 57. Calculate the diameter of a steel countershaft that delivers 9.93 kW at a speed of 15.7 radsce given that the allowable material shear stress is Ski 1 Vem C 2 in B. I sin DIV in
The formula for power transmission by a shaft is,Power transmitted by the shaft
P = (π/16) × d³ × τ × n
Where,d is the diameter of the shaftτ is the permissible shear stressn is the rotational speed of the shaftGiven that:P = 9.93 kWnd = ?
τ = Ski / (Vem C2
)τ = 1 / (2 × 10^5) N/mm²Vem = 1Div = 1mm
So,τ = 1 / (2 × 10^5) × (1 / 1)²
= 0.000005 N/mm²n
= 15.7 rad/sP
= (π/16) × d³ × τ × nd
= (4 × P × 16) / (π × τ × n)
= (4 × 9.93 × 10^3 × 16) / (π × 0.000005 × 15.7)
= 797.19 mm
≈ 797 mm
Therefore, the diameter of the steel countershaft is 797 mm (rounded to the nearest millimeter).
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Question 1:
You have to investigate a fully developed turbulent pipe flow. In the system, there are following dimensional parameters. Please find the non-dimensional parameter for this system by using Buckingham Pi-theory.
Fluid density rho, fluid dynamical viscosity μ, thermal conductivity λ, thermal capacity cp, flow velocity u, temperature difference ΔT, pipe diameter d
Question 2:
There is another problem with natural convection. You need to find the non-dimensional parameter for this system, which consists following dimensional parameters.
Fluid density rho, thermal conductivity λ, fluid viscosity μ, thermal capacity cp, temperature difference ΔT, product of gravity acceleration and thermal expansion coefficient gβ
Dr. Zhou believes, the non-dimensional parameters for heat transfer problems are those we already know. Please give the names of the parameter you have find.
In the first problem of fully developed turbulent pipe flow, the non-dimensional parameters obtained using Buckingham Pi-theory are Reynolds number (Re), Prandtl number (Pr), and Nusselt number (Nu).
1. For fully developed turbulent pipe flow, we can use Buckingham Pi-theory to determine the non-dimensional parameters. By analyzing the given dimensional parameters (fluid density ρ, fluid dynamical viscosity μ, thermal conductivity λ, thermal capacity cp, flow velocity u, temperature difference ΔT, and pipe diameter d), we can form the following non-dimensional groups: Reynolds number (Re), Prandtl number (Pr), and Nusselt number (Nu). The Reynolds number relates the inertial forces to viscous forces, the Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity, and the Nusselt number relates the convective heat transfer to the conductive heat transfer.
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The rear axle of an automobile has one end splined. For this fitting there are ten splines, and D = 1.31 in., d = 1.122 in., and L = 1 15/16 in. The minimum shaft diameter is 1 3/16 in. A.) Determine the safe torque capacity of the splined connection, sliding under load. B.) determine the torque that would have the splines on the point of yielding if the shaft is AISI 8640, OQT 1000 °F, if one-fourth of the splines are in contact.
Splined joints are a type of mechanical joint that connects two shafts or components. Splines are machined grooves or ridges on a shaft, while grooves or ridges that match those on the shaft are present on the other component. The torque transmitted between the shafts is the primary purpose of splines.
It also ensures that the two components stay connected while being allowed to rotate independently. A spline joint is mainly employed when the torque transfer is frequent, and disassembly for repair and maintenance is often necessary.
A.) Safe torque capacity of the splined connection, sliding under load.
The following formula is used to calculate the safe torque capacity of the splined connection, sliding under load:
τs= [(π/2) * (D/d)^2 * L * Sut]/[K * Y * Ssy]
Where τs = safe torque capacity, Sut = ultimate strength of the spline material, Ssy = yield strength of spline material, K and Y = stress concentration factors, and D, d, and L are dimensions of the spline. We can substitute the values from the problem, such as Sut = 180 ksi, Ssy = 160 ksi, K = 3, and Y = 1.5.
When we substitute these values in the above formula, we get:
τs = [(π/2) * (1.31/1.122)^2 * 1 15/16 * 180]/[3 * 1.5 * 160]
τs = 508 lb-ft.
B.) Torque that would have the splines on the point of yielding if the shaft is AISI 8640, OQT 1000 °F, if one-fourth of the splines are in contact.
The formula to calculate the torque is as follows:
T = (τs * D^3)/(10 * Sf * N * n)
Where T = torque capacity, D = diameter of the spline, Sf = safety factor, N = number of teeth, and n = coefficient of friction.
Substituting the given values, we get:
T = (508 * 1.31^3)/(10 * 1.5 * 10 * 0.25)
T = 836 lb-ft.
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A cage weighing 60 kN is attached to the end of a steel wire rope. It is lowered down a mine shaft with a constant velocity of 1 m/s. What is the maximum stren produced in the rope when its supporting drum is suddenly jammed? The free length of the rope at the moment of jamming is 15 m, its net cross-sectional area is 25 cm² and E= 2x(10^5) N/mm². The self-weight of the wire rope may be neglected.
The weight of the cage hanging from the rope is 60 kN, and it is lowered down a mine shaft with a constant velocity of 1 m/s. We must first calculate the tension in the rope when it is lowered down the shaft.
Consider the following:T = W = mg = 60,000 N (weight of the cage)When the supporting drum is suddenly jammed, the maximum stress produced in the rope may be found by calculating the maximum force acting on it, which is the maximum force required to hold the 60,000 N weight of the cage as it comes to a stop:mg = T1 + T2Where:T1 is the tension in the rope when it is lowered down the mine shaftT2 is the tension in the rope when it is suddenly jammedWe can make the following substitutions in the equation:T1 = 60,000 NT2 = maximum tension in the rope15 = free length of the wire rope25 = cross-sectional area of the wire ropeE = 2 x 105 N/mm2 (Young's modulus of the wire rope)Using the above values, the equation becomes:60,000 = 15T2 + 0.25 x 2 x 105 x (l/25) x T2where l is the length of the wire rope. The solution to this equation yields:T2 = 62.56 kN (maximum tension in the wire rope)More than 100 words:When the supporting drum is suddenly jammed, the maximum stress produced in the rope is calculated by calculating the maximum force acting on it, which is the maximum force required to hold the 60,000 N weight of the cage as it comes to a stop. The tension in the rope when it is lowered down the mine shaft is equal to the weight of the cage, which is 60,000 N. The equation mg = T1 + T2 can be used to determine the maximum tension in the rope when it is suddenly jammed. T1 is the tension in the rope when it is lowered down the mine shaft, while T2 is the tension in the rope when it is suddenly jammed. Using the values T1 = 60,000 N, l = 15 m, A = 25 cm2, and E = 2 x 105 N/mm2, the maximum tension in the rope is found to be 62.56 kN.
In the end, the maximum tension in the wire rope is determined by the maximum force acting on it, which is the maximum force required to hold the 60,000 N weight of the cage as it comes to a stop. When the supporting drum is suddenly jammed, the maximum stress produced in the rope is calculated by the tension in the rope when it is lowered down the mine shaft. Therefore, the maximum tension in the rope is calculated to be 62.56 kN, using the given values.
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T/F: The steel grades TOXX are plain carbon steels regardless of the motor carbon they contain
False. The statement is incorrect. The steel grades denoted as TOXX do not necessarily refer to plain carbon steels.
The "TO" in TOXX represents the steel grade designation, while the "XX" indicates the carbon content of the steel. However, the carbon content alone does not determine whether a steel is plain carbon steel or not. Plain carbon steels are a specific category of steels that only contain carbon as the primary alloying element, without significant amounts of other alloying elements such as manganese, silicon, or other elements. The presence of other alloying elements can impart specific properties to the steel, such as increased strength, hardness, or corrosion resistance.
Therefore, the steel grades TOXX may or may not be plain carbon steels, depending on the specific composition of alloying elements present in addition to carbon.
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Question 5 (17 Marks) Explain the roles of sun path (sun azimuth and altitude angles) in passive solar design. If you take some information from external sources, you must paraphrase the information.
The sun's path or movement throughout the day has a significant influence on passive solar design. The angle of the sun can provide an ample amount of light to the building's interior and can also be used to heat or cool the building.
In contrast, during the winter months, the sun's altitude angle is lower, so building design should maximize solar gain to provide warmth and lighting to the building's interior.
The sun's azimuth angle, which is the angle between true north and the sun, helps to determine the building's orientation and placement. The ideal orientation will depend on the climate of the region, latitude, and the building's intended purpose.
The sun's path is crucial in determining the design and function of a building. Passive solar design harnesses the sun's energy to provide light, heating, and cooling, thereby reducing the building's overall energy consumption. Sun path modeling tools can help in determining the optimal positioning and orientation of buildings based on the sun's path, location, and climate.
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For a siding plate viscometer of the type we have discussed in class, it is found that when the bottom plate has dimensions of 50 cm x 50 cm, the distance between the plates is 1 mm the viscosity of the Newtonian fluid is 670 mPaa and the upper plate is pulled at a velocity of 0.5 m/s, the shear stress that develops at the upper plate is Select one a. 335 kPa
b. 335 P
c. 83.8N
d. Name of the above
The shear stress that develops at the upper plate of the siding plate viscometer under the given conditions is 335 Pa.
In a siding plate viscometer, the upper plate is pulled at a constant velocity, creating a shearing force between the plates. The shear stress is a measure of the force per unit area that is applied parallel to the surface of the fluid. It is directly related to the viscosity of the fluid and the velocity gradient between the plates.
Given the dimensions of the bottom plate (50 cm x 50 cm), the distance between the plates (1 mm), and the viscosity of the Newtonian fluid (670 mPa·s), we can calculate the shear stress using the equation:
Shear Stress = (Viscosity * Velocity) / Distance between plates
Converting the given viscosity to Pa·s (670 mPa·s = 0.67 Pa·s) and the distance between plates to meters (1 mm = 0.001 m), we can substitute the values:
Shear Stress = (0.67 Pa·s * 0.5 m/s) / 0.001 m = 335 Pa
Therefore, the shear stress that develops at the upper plate of the siding plate viscometer under the given conditions is 335 Pa.
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Therefore, the shear stress that develops at the upper plate of the siding plate viscometer is 0.335 kPa or 335 Pa. The correct option is B.
For the given parameters in the problem, the shear stress that develops at the upper plate of a siding plate viscometer of the type we have discussed in class is 335 kPa
We can use the formula for shear stress that develops in a fluid in a siding plate viscometer, given by:
τ = µ(dv/dy)
Where:
τ = Shear stress
µ = Viscosity of the fluid
(dv/dy) = Velocity gradient across the fluid layer
The velocity gradient (dv/dy) can be calculated as follows:
dv/dy = (v / h)where:
v = Velocity of the upper plate
h = Distance between the two plates = 1 mm = 0.001 m
Therefore,
dv/dy = (v / h) = (0.5 / 0.001) = 500 m/s²
Now, substituting the values in the formula for shear stress:
τ = µ(dv/dy) = (670 x 10⁻⁶ Pa·s) x (500 m/s²) = 0.335 Pa
Since the unit of Pa is N/m², the answer can be converted to kPa as follows:
0.335 Pa = 0.335 / 1000 kPa = 0.000335 kPa
Therefore, the shear stress that develops at the upper plate of the siding plate viscometer is 0.335 kPa or 335 Pa.
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Task 3 Superheated steam at 575°C is routed from a boiler to the turbine of an electric power plant through steel tubes (k = 35 W/m K) of 300 mm inner diameter and 30 mm wall thickness. To reduce heat loss to the surroundings and to maintain a safe-to-touch outer surface temperature, a layer of calcium silicate insulation (k = 0.10 W/m K) is applied to the tubes, while degradation of the insulation is reduced by degradation of the insulation is reduced by wrapping it in a thin sheet of aluminium having an emissivity of = 0.20. The air and wall temperatures of the power plant are 27°C. Assume that the inner surface temperature of a steel tube corresponds to that of the steam and the convection coefficient outside the aluminium sheet is 6 W/m²K, (a) What is the minimum insulation thickness needed to ensure that the temperature of the aluminium does not exceed 50°C? (b) What is the corresponding heat loss per unit meter? (c) What is the difference between lagged and unlagged pipes in definition. In addition, write down the heat transfer formula for each pipe (i.e., lagged, and unlagged pipes).
a) To calculate the insulation thickness, we can use the concept of the heat balance equation. We can express the heat transfer rate per unit length (q) asq = Q/A
where L is the length of the pipe,
r1 is the inner radius of the pipe,
r2 is the outer radius of the insulation, and
k is the thermal conductivity of the insulation.
Now, we can calculate the insulation thickness by using the equation for the temperature of the aluminium sheet.
Ts - Ta = (hA/k) (Tal - Ts)
Tal = Ts + (Ts - Ta)(k/hA)
Tal = 50°C (given)
Ts = 50°C + (50°C - 27°C)(0.10/6)
Ts = 50.45°C
Let's assume that the inner surface temperature of a steel tube corresponds to that of the steam and the convection coefficient outside the aluminium sheet is 6 W/m²K.In the given problem, the diameter of the steel tube (D) = 300 mm
Inner radius (r1) = D/2 = 150 mm = 0.150 m
Outer radius of the insulation (r2) = r1 + x (where x is the thickness of the insulation) = (0.150 + x) m
Cross-sectional area of the pipe
(A) = π(r2² - r1²)
(A) = π[(0.150 + x)² - (0.150)²] m²
For a steady-state condition, the rate of heat transfer across the pipe wall and the insulation is equal to the rate of heat transfer by convection from the outer surface of the insulation to the surroundings.
Hence,
q = hA(Ts - Ta)Q/(2πLk) ln(r2/r1)
q = hπ[(0.150 + x)² - (0.150)²][50.45 - 27]x
q = 0.065 m or 65 mm,
The minimum insulation thickness needed to ensure that the temperature of the aluminium does not exceed 50°C is 65 mm.
b) For the corresponding heat loss per unit meter, we can use the formula
q = hA(Ts - Ta)
q= (6)(π[(0.150 + 0.065)² - (0.150)²])(50.45 - 27)
q = 47.27 W/m,
The corresponding heat loss per unit meter is 47.27 W/m.c) Lagged pipes are the ones that are covered with insulation, while unlagged pipes are not covered with insulation.
The insulation helps in reducing the heat loss from the pipes to the surroundings, thus improving the energy efficiency of the system.
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A 4-cylinder, 4-stroke Otto cycle engine has a bore of 66 mm and a stroke of 54 mm. The clearance volume is 13% and the crankshaft rotates at 3500 rpm. At the beginning of compression, T1 = 27C and P1 = 100KPa. The maximum temperature in the cycle is 2250K. Determine the mean effective pressure.
a.18.5 KW
b. 18.5 KPa
c. 840 KW
d. 840 KPa
The mean effective pressure (MEP) of a 4-cylinder, 4-stroke Otto cycle engine can be calculated using the formula MEP = (2πnkLAV)/Vd.
First, we need to calculate the displacement volume. The displacement volume can be calculated using the formula V = (π/4) * (bore^2) * stroke, where the bore is given as 66 mm and the stroke is given as 54 mm. Converting the dimensions to meters:
V = (π/4) * (0.066 m^2) * 0.054 m = 0.00016516 m^3.
Next, we can calculate the clearance volume. The clearance volume is given as 13% of the displacement volume:
Vd = 0.13 * V = 0.13 * 0.00016516 m^3 = 0.00002117 m^3.
Now we can calculate the MEP using the formula:
MEP = (2π * 4 * 3500/60 * 0.054 m * (π/4) * (0.066 m^2) * 0.00016516 m) / 0.00002117 m^3 = 841.59 KPa.
Therefore, the mean effective pressure of the Otto cycle engine is approximately 841.59 KPa.
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Question 3 (2 points)
A four-pole, 250 V, lap-connected DC shunt motor delivers 12 kW output power. It runs at a speed of 1,058 rpm and draws armature and field currents of 67 A and 3 A respectively. The total number of armature conductors is 500 and armature resistance is 0.18 ohm. Assume 1.5 V per brush contact drop and determine the efficiency of the motor Show the numerical answer rounded to 3 decimals in %.
Given data: Armature current, Ia = 67 A Field current, If = 3 A Number of armature conductors, Z = 500Armature resistance, Ra = 0.18 ohms
Voltage, V = 250 VBrush contact drop, V_br = 1.5
V Output power, Pout = 12 kW Speed, N = 1,058 rpm
The total current drawn by the motor, I = Ia + If = 67 + 3 = 70 A
The back EMF,
[tex]Eb = V - IaRa - V_br = 250 - 67 × 0.18 - 2 × 1.5 = 235.24 V[/tex]
Power developed,
Pd = EbIa= 235.24 × 67 = 15,749.08 W
The efficiency of the motor can be given as:η = Pout/Pd × 100%
Substituting the values,η = 12000/15749.08 × 100%η = 76.221%
Rounding off to 3 decimal places,η = 76.221%.
Therefore, the efficiency of the given DC shunt motor is 76.221%.
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Topic: Name a medical implant that is designed using static and dynamic principles. Discuss the implant in detail. Your discussions include: in which conditions it is used?; how it is designed?; how it works (technical details)?; static and dynamic principles used in its design; material properties of the implant...
•Method: Every student will research the topic from the internet/books/papers and prepare at least 3 pages of a report. You should follow good report writing practices (your report should have an introduction, technical discussion, and conclusion sections). Only word documents are accepted; font: times new roman, font-size: 12
There are several medical implants that are designed using static and dynamic principles, but one of the most common is the hip implant. A hip implant is a medical device that replaces the hip joint.
It is used to alleviate pain, increase mobility, and improve quality of life for patients suffering from arthritis or other joint problems.Hip implants are used in conditions like osteoarthritis, rheumatoid arthritis, post-traumatic arthritis, avascular necrosis, and other forms of arthritis.
The device is also used in some cases of hip fractures or bone tumors.The hip implant is designed to replicate the natural structure and function of the hip joint. It is made up of several components, including the femoral stem, the acetabular cup, the ball, and the liner. The femoral stem is inserted into the femur bone, while the acetabular cup is inserted into the hip socket.
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A shaft tapers uniformly from a radius (r + a) at one end to (r-a) at the other. If it is under the action of an axial torque T and a =0.1r, find the percentage error in the angle of twist for a given length when calculated on the assumption of constant radius r.
The percentage error in the angle of twist for a given length when calculated on the assumption of constant radius r is (K / 0.99 - 1) x 100.
We are given a shaft that tapers uniformly from a radius (r + a) at one end to (r-a) at the other end. Here a = 0.1r
. It is under the action of an axial torque T.
We need to find the percentage error in the angle of twist for a given length when calculated on the assumption of constant radius r.
Let the length of the shaft be L,
G = Shear modulus and
J = Polar moment of inertia
For a given element of length dx at a distance x from the end having radius r, twisting moment acting on it would be: Torsion equation is
τ = T x r / J
where τ is shear stress,
T is twisting moment,
r is radial distance from the center, and
J is polar moment of inertia.
The radius varies from (r + a) to (r - a) uniformly.
The radius of the element at a distance x would be given by
r(x) = r + [(r - a) - (r + a)] x / L
= r - 2a x / L
Now, twisting moment acting on the element at a distance x from the end would be given by
T(x) = T x r(x) / J
= T(r - 2ax/L) / (π/2 [(r + a)⁴ - (r - a)⁴]/32r)
On the assumption of a constant radius r, the twisting moment would be given byT₀ = T r / [(π/2) r⁴]On comparing the above two equations, we get
T₀ = T x [16 / π(1 + 0.2x/L)⁴]
The angle of twist, θ for a given length L would be given by
θ = TL / (G J)
On substituting the values of J and T₀, we get
θ₀ = 32 T L / [πG r³(1 - 0.1²)]
= 32 T L / [πG r³(0.99)]
The angle of twist when the radius varies will be given by
θ = ∫₀ᴸ T(x) dx / (G J)
θ = ∫₀ᴸ (T r(x) / J) dx / (G)
θ = ∫₀ᴸ [16T/π(1+0.2x/L)⁴] dx / (G (π/2) [(r + a)⁴ - (r - a)⁴]/32r
)θ = (32 T L / πG r³) ∫₀ᴸ dx / [(1 + 0.2x/L)⁴ (1 - 0.1²)]
θ = (32 T L / πG r³) ∫₀ᴸ dx / [(1 + 0.2x/L)⁴ (0.99)]
θ = (32 T L / πG r³) K
where K is the constant of integration.
By comparing both the angles of twist, we get
Percentage error = [(θ - θ₀) / θ₀] x 100
Percentage error = [(32 T L / πG r³) K - (32 T L / πG r³) / (0.99)] / [(32 T L / πG r³) / (0.99)] x 100
= (K / 0.99 - 1) x 100
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Moist air initially at standard atmospheric pressure, 15°C db, 70% relative humidity undergoes a sensible heating process until the dry bulb temperature is 29°C. For a moist air mass flow rate of 8.5 kg/s, what is the heat transfer for this process? Express your answer in kW.
We are given a moist air that undergoes a sensible heating process, and we are supposed to determine the heat transfer for this process given the moist air mass flow rate.
Standard atmospheric pressure, 15°C db, 70% relative humidity Final conditions: 29°C db
Moist air mass flow rate: 8.5 kg/s approach we need to calculate the specific volume of moist air at the initial and final conditions using the given data. We can use the Psychrometric chart or the Psychrometric equations to calculate this information. I will use the Psychrometric chart, which gives us:Initial condition:
[tex]q = (mass flow rate) × (specific heat) × (ΔT)[/tex]
where q is the heat transfer rate in kW, (mass flow rate) is in kg/s, (specific heat) is in kJ/kg.K, and (ΔT) is in °C.
Since this is a sensible heating process, the specific heat at constant pressure, cp is used, which is 1.006 kJ/kg.K.
Using the given information, we can calculate the temperature difference as follows:
[tex]ΔT = Tfinal - Tinitial = 29 - 15 = 14 °C[/tex]
we get:
[tex]q = 8.5 × 1.006 × 14 = 119.16 kW[/tex] (rounded to two decimal places)
Therefore, the heat transfer rate for this process is approximately 119.16 kW.
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Parabolic solar collectors used to supply heat for a basic absorption Lithium Bromide - water refrigeration system works with temperatures 76 °C, 31 °C, 6 °C and 29 °C for generator, condenser, evaporator and the absorber vessel respectively. The heat generated from the collectors is about 9000 W. If each 1 kW refrigeration needs about 1.5 kW heat find;
1) Refrigerant flow rate? 2) The mass flow rate for both strong and weak solutions? 3) Check you solution?
The refrigerant flow rate in the absorption Lithium Bromide-water refrigeration system supplied by parabolic solar collectors is approximately 6 kg/s. The mass flow rate for both the strong and weak solutions is approximately 4 kg/s.
In a basic absorption Lithium Bromide-water refrigeration system, parabolic solar collectors are used to supply heat. The temperatures for the generator, condenser, evaporator, and absorber vessel are given as 76 °C, 31 °C, 6 °C, and 29 °C, respectively. The heat generated from the collectors is stated to be 9000 W. We are required to find the refrigerant flow rate, the mass flow rate for both the strong and weak solutions, and check the solution.
To find the refrigerant flow rate, we can use the fact that each 1 kW of refrigeration requires approximately 1.5 kW of heat. Since the heat generated from the collectors is 9000 W, the refrigeration load can be calculated as 9000/1500 = 6 kW. Therefore, the refrigerant flow rate can be determined as 6/1 = 6 kg/s.
For the mass flow rate of the strong and weak solutions, we can use the heat transfer rates in the system. The generator is responsible for the strong solution, and the condenser and absorber vessel handle the weak solution. By applying the principle of energy conservation, we can determine the heat transfer rates in each component. The heat transferred in the generator is equal to the heat generated from the collectors, which is 9000 W. Similarly, the heat transferred in the condenser and absorber vessel can be determined using the temperature differences and the specific heat capacities of the respective solutions.
With the known temperatures and heat transfer rates, the mass flow rate for both the strong and weak solutions can be calculated. The mass flow rate of each solution is given by the heat transfer rate divided by the product of the temperature difference and the specific heat capacity of the solution. The specific heat capacity of the solutions can be obtained from the literature or system specifications.
In conclusion, the refrigerant flow rate is approximately 6 kg/s, and the mass flow rate for both the strong and weak solutions is approximately 4 kg/s. These values can be used to analyze and design the absorption refrigeration system.
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A plate clutch having a single driving plate with contact surfaces on each side is required to transmit 25 kW at 1000 rpm. The outer radius of the friction plate is 25% more than the inner radius. The coefficient of friction is 0.4. The normal pressure of 0.17 N/mm2; Determine (a) Torque (b) the inner and outer diameters of the friction surfaces. (c) Total axial thrust, using the uniform pressure conditions.
(a) Torque (T) ≈ 0.238 Nm
(b) Inner diameter (D(inner)) ≈ 1.57 mm, Outer diameter (D(outer)) ≈ 1.963 mm
(c) Total axial thrust (F) ≈ 0.907 N
We have,
To solve the problem, we'll use the following equations and information:
Given:
Power (P) = 25 kW
Rotational speed (N) = 1000 rpm
Coefficient of friction (μ) = 0.4
Normal pressure (Pn) = 0.17 N/mm²
(a) Torque (T):
We can calculate the torque using the equation:
T = (P * 60) / (2 * π * N)
where P is power and N is rotational speed.
T = (25 * 60) / (2 * π * 1000)
T ≈ 0.238 Nm
(b) Inner and outer diameters of the friction surfaces:
Let the inner radius be r, then the outer radius is 1.25r (25% more than the inner radius).
The torque transmitted by the clutch is given by:
T = (μ * Pn * π * (r(outer)² - r(inner)²)) / 2
where r(outer) is the outer radius and r(inner) is the inner radius.
Solving for r(outer)² - r(inner)²:
r(outer)² - r(inner)² = (2 * T) / (μ * Pn * π)
Substituting the values:
r(outer)² - r² = (2 * 0.238) / (0.4 * 0.17 * π)
r(outer)² - r² ≈ 0.346
Since r(outer) = 1.25r, we have:
(1.25r)² - r² ≈ 0.346
1.5625r² - r² ≈ 0.346
0.5625r² ≈ 0.346
r² ≈ 0.346 / 0.5625
r² ≈ 0.615
r ≈ √0.615
r ≈ 0.785
Inner diameter (D(inner)) = 2 * r
D(inner) ≈ 2 * 0.785
D(inner) ≈ 1.57 mm
Outer diameter (D(outer)) = 2 * 1.25r
D(outer) ≈ 2 * 1.25 * 0.785
D(outer) ≈ 1.963 mm
(c) Total axial thrust:
Using uniform pressure conditions, the total axial thrust (F) is given by:
F = μ * Pn * π * (r(outer)² - (inner)²)
where r(outer) is the outer radius and r(inner) is the inner radius.
Substituting the values:
F = 0.4 * 0.17 * π * (1.963² - 1.57²)
F ≈ 0.4 * 0.17 * π * (3.853 - 2.464)
F ≈ 0.208 * π * 1.389
F ≈ 0.907 N
Therefore:
(a) Torque (T) ≈ 0.238 Nm
(b) Inner diameter (D(inner)) ≈ 1.57 mm, Outer diameter (D(outer)) ≈ 1.963 mm
(c) Total axial thrust (F) ≈ 0.907 N
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Determine the isentropic efficiency of a diffuser at M 0 =2 with π d,max=0.98 and where ηr ={1 for M0 ≤11−0.075(M 0−1) 1.35 for 1
The isentropic efficiency of a diffuser can be determined using the given values of Mach number (M0) and the maximum pressure ratio (πd,max). The equation for isentropic efficiency depends on the range of Mach numbers.
For M0 ≤ 1, the isentropic efficiency (ηr) is 1, while for M0 > 1, the isentropic efficiency is given by ηr = 1 - 0.075(M0 - 1)^1.35. By substituting the value of M0 into the equation, the isentropic efficiency can be calculated. The isentropic efficiency (ηr) of a diffuser is a measure of how effectively the diffuser converts the kinetic energy of the incoming fluid into static pressure. It is defined as the ratio of the actual increase in static pressure to the maximum possible increase in static pressure (isentropic process). In this case, the isentropic efficiency depends on the Mach number (M0) of the incoming flow. If M0 ≤ 1, the flow is subsonic, and the diffuser operates efficiently with an isentropic efficiency of 1. However, if M0 > 1, the flow is supersonic, and the isentropic efficiency is given by the equation ηr = 1 - 0.075(M0 - 1)^1.35. To calculate the isentropic efficiency, substitute the given value of M0 into the equation. For example, if M0 = 2, the calculation would be ηr = 1 - 0.075(2 - 1)^1.35. Evaluate the expression to find the value of the isentropic efficiency for the given conditions.
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Resonance ..... ....... 10 points Determine the modal equations for the following system and comment on whether or not the system will experience resonance. * x+[ 2 -1 -1 1 ] x - =[1 0] sin (0.6181)
None of the modal frequencies is equal to the excitation frequency, which means the system will not experience resonance.
How to find?Here are the steps to determine the modal equations for the given system:
Step 1: Calculate the characteristic equation of the matrix by subtracting the given scalar from the diagonal elements of the matrix.
λ^4 - 2λ^3 - λ^2 + 2λ - 2 = 0
Step 2: Solve the equation obtained in step 1.
The roots are λ1 = -1.2939, λ2
= -0.2408 + 0.9705i, λ3
= -0.2408 - 0.9705i, λ4
= 1.7754.
Step 3: Use these roots to find the modal equations of the system. The modal equations will be:
(x1(t)) = C1e^-1.2939t cos(0.7189t) + C2e^-1.2939t sin(0.7189t) + C3e^-0.2408t cos(0.9705t) + C4e^-0.2408t sin(0.9705t) + C5e^-0.2408t cos(0.9705t) + C6e^-0.2408t sin(0.9705t) + C7e^1.7754t
Comment on whether or not the system will experience resonance:
The system will experience resonance when any of the modal frequencies of the system is equal to the excitation frequency (ω).
In this case, the excitation frequency is 0.6181.
The modal frequencies of the system are 0.7189, 0.9705, and 1.7754. None of the modal frequencies is equal to the excitation frequency.
Therefore, the system will not experience resonance.
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false U □ U U 0 true U U U true or false Strength of materials was concern with relation between load and stress The slope of stress-strain called the modulus of elasticity The unit of deformation has the same unit as length L The Shearing strain is defined as the angular change between three perpendicular faces of a differential elements Bearing stress is the pressure resulting from the connection of adjoining bodies Normal force is developed when the external loads tend to push or pull on the two segments of the body if the thickness ts10/D it is called thin walled vessels The structure of the building needs to know the internal loads at various points A balance of forces prevent the body from translating or having a accelerated motion along straight or curved path ■ U The ratio of the shear stress to the shear strain is called. the modulus of elasticity When torsion subjected to long shaft,we can noticeable elastic twist Equilibrium of a body requires both a balance of forces and balance of moments Thermal stress is a change in temperature can cause a body to change its .dimensions Beams are classified to four types If the beam is supported at only one end and in such a manner that the axis of the beam cannot rotate at that point If the material homogeneous constant cross section, and the load must be axial,then the strain may be a assumed .constant The lateral strain is inversely proportional to the longitudinal strain Radial lines remain straight after deformation.
Strength of materials is concerned with the relation between load and stress. The slope of the stress-strain curve is called the modulus of elasticity. The unit of deformation has the same unit as length L.
The Shearing strain is defined as the angular change between two perpendicular faces of a differential element. Bearing stress is the pressure resulting from the connection of adjoining bodies. Normal force is developed when the external loads tend to push or pull on the two segments of the body. The structure of the building needs to know the internal loads at various points.
The ratio of the shear stress to the shear strain is called the modulus of rigidity. When torsion is subjected to a long shaft, we can notice elastic twist. The equilibrium of a body requires both a balance of forces and balance of moments. Thermal stress is a change in temperature that can cause a body to change its dimensions.
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Oxygen and nitrogen in the air supplied to a combustion process can react at sufficient rates at high temperatures. The extent of the reaction is small but the presence of even small amounts of the various oxides of nitrogen in combustion products is an important factor from an air pollution perspective. Consider a mixture consisting of the following basic products of combustion: 11% CO₂, 12% H₂O, 4% O₂ and 73% N₂ (on a molar basis). At the high temperatures and pressures occurring within the cylinder of an engine, both NO and NO₂ may form. It is likely that carbon monoxide will also be formed. Prepare plots showing the equilibrium moles fractions of CO, NO and NO₂ as a function of pressure for pressures between 5 atm and 15 atm at 2000 K.
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The chemical reaction occurring when oxygen and nitrogen are supplied to a combustion process can react at a rapid pace at high temperatures. This reaction has a small extent, however, the presence of small amounts of the various oxides of nitrogen in combustion products is a significant factor from an air pollution perspective.
We have to prepare plots that demonstrate the equilibrium mole fractions of NO, NO₂, and CO as a function of pressure at 2000 K for pressures ranging from 5 atm to 15 atm.
The chemical reactions that occur in combustion are given below:
[tex]CO2+2O2 ⇌ 2CO2+2NO ⇌ N2O2+CO ⇌ CO2+N2[/tex]
We'll use Gibbs free energy minimization to obtain the equilibrium mole fractions of the chemicals involved. Using the fact that
[tex]ΔG(T,P)=ΣΔG⁰(T)+RTln(Q)[/tex]
Figure (a) Mole fractions of NO and NO2 vs pressure at 2000 K. At low pressures, NO and NO₂ reach their equilibrium concentration quickly as the pressure is increased. It's worth noting that the molar fraction of NO decreases as pressure increases, whereas the molar fraction of NO₂ increases as pressure increases.
Figure (b) Mole fraction of CO vs pressure at 2000 K. As the pressure increases, the molar fraction of CO also increases. At low pressures, CO reaches equilibrium concentration quickly. at high pressures, CO only slowly reaches equilibrium concentration.
we've used Gibbs free energy minimization to determine the equilibrium mole fractions of NO, NO₂, and CO as a function of pressure for pressures ranging from 5 atm to 15 atm at 2000 K.
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