The final temperature of the gas in the container after releasing half of it is approximately 13.65°C. To solve this problem, we can use the ideal gas law equation:
PV = mRT
where P is thThe final temperature of the gas in the container after releasing half of it is approximately 13.65°C. To solve this problem, we can use the e pressure, V is the volume, m is the mass, R is the gas constant, and T is the temperature.
Given:
Initial pressure (P1) = 300 kPa
Initial temperature (T1) = 50°C = 50 + 273.15 K
Final pressure (P2) = 220 kPa
Mass of the gas (m) = 3 kg
First, let's calculate the initial volume (V1) of the gas using the ideal gas law:
V1 = (mRT1) / P1
Next, we can calculate the final mass of the gas remaining in the container after releasing half of it:
Final mass = 0.5 * Initial mass
Final mass = 0.5 * 3 kg = 1.5 kg
Now, we can calculate the final volume (V2) of the gas using the ideal gas law and the final mass:
V2 = (mRT2) / P2
Since the initial and final volumes are the same (as it is a rigid container), we can equate V1 and V2:
(mRT1) / P1 = (mRT2) / P2
We can cancel out the mass (m) and the gas constant (R) from both sides of the equation:
T1 / P1 = T2 / P2
Now, we can rearrange the equation to solve for the final temperature (T2):
T2 = (T1 * P2) / P1
Substituting the given values:
T2 = (50 + 273.15) * 220 / 300
Calculating the final temperature:
T2 ≈ 286.80 K
Converting the temperature to °C:
T2 ≈ 13.65°C
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. Choose the correct statement a C a. Nozzle velocity is known as the mean velocity b. Impact velocity is related to the impulsive force c. d. the total weight added for all trials regardless of the vane shape The flowrate in the impact of jet experiment is measured in mm^2/s In
The correct statement among the given options is, "Impact velocity is related to the impulsive force". The Impact of Jet apparatus is an experimental setup that shows the force developed by a jet of fluid striking a plane or a curved plate. The experiment is significant in mechanical engineering as it helps in determining the impact force exerted by a jet of water or a fluid on various vanes.
The velocity of a fluid jet from a nozzle produces an impact force on any surface it strikes. The force developed by the jet is a function of the fluid velocity and density. The following equation describes the force developed by a fluid jet:
Force = density × Velocity × Area.
From the equation, it can be said that the force is proportional to the velocity of the fluid jet. The greater the velocity, the greater the force developed. Hence, impact velocity is related to the impulsive force. The flow rate in the Impact of Jet experiment is measured in m³/s.
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A 1.92-KV, 1100-HP, unity power factor, 60-Hz, 2-pole, Δ-connected synchronous motor has a synchronous reactance of 10.1 Ω per-phase and a negligible armature resistance. The friction and windage losses together with the core losses are 4.4 KW. The open-circuit characteristic of this motor is shown below in a tabular form This motor is being supplied from a large power system.
A synchronous motor is a type of AC motor that o corresponding to the frequency of the applied voltage. The output power of a synchronous motor is proportional to the power supply voltage and the synchronous reactance of the motor.
If the supply voltage is held constant, reactance.The given synchronous motor has a rating of 1.92 kV, 1100 HP, and unity power factor. It is 60-Hz, 2-pole, and delta-connected. The synchronous reactance of the motor is 10.1 Ω per-phase. Additionally, the motor's armature resistance is negligible.
The friction and losses combined with the core losses are 4.4 kW. The open-circuit characteristic of the motor is tabulated below in detail:Exciting current 5.5 A
Field voltage (volts) 25.6
Armature current (amperes) 167.0
Power factor 0.86 lagging.
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A gas goes over the cycle ABCA where AC is an isotherm and AB is an isobar. the volume at B and A are 2 L and 8L respectively. L=10-3m³
Assume PV= Constant and find the followings:
a. Sketch the PV diagram of the process (5pts)
b. The pressure at point C. (10 pts)
C. the work done in part C-A of the cycle (15 pts)
d. the heat absorbed or rejected in the full cycle (10 pts)
a. Sketching the PV diagram of the process:
In the PV diagram, the x-axis represents volume (V) and the y-axis represents pressure (P).
Given:
Volume at point B (VB) = 2 L
Volume at point A (VA) = 8 L
We know that PV = constant for the process.
The PV diagram for the cycle ABCA will be as follows:
A
______|______
| |
| C |
| |
|_____________|
B
b. The pressure at point C:
Since AC is an isotherm and AB is an isobar, we can use the ideal gas law to determine the pressure at point C.
PV = constant
At point A: P_A * V_A = constant
At point C: P_C * V_C = constant
Since the volume at point C is not given, we need more information to determine the pressure at point C.
c. The work done in part C-A of the cycle:
To calculate the work done in part C-A of the cycle, we need to know the pressure and volume at point C. Without this information, we cannot determine the work done.
d. The heat absorbed or rejected in the full cycle:
The heat absorbed or rejected in the full cycle can be calculated using the First Law of Thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat (Q) absorbed or rejected by the system minus the work (W) done on or by the system.
ΔU = Q - W
Without the specific values of heat or additional information about the process, we cannot calculate the heat absorbed or rejected in the full cycle.
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Determine the range of K for stability of a unity feedback control system whose open-loop transfer function is K G(s) = K/s(s+ 1)(s + 2)
The range of K for stability of the given control system is $0 < K < 6$. Therefore, the answer is : Range of K for stability of a unity feedback control system whose open-loop transfer function is K G(s) = K/s(s+ 1)(s + 2) is 0 < K < 6.
Given Open loop transfer function: [tex]$$K G(s) = \frac{K}{s(s+ 1)(s + 2)}$$[/tex]
The closed-loop transfer function is given by: [tex]$$\frac{C(s)}{R(s)} = \frac{KG(s)}{1 + KG(s)}$$$$= \frac{K/s(s+ 1)(s + 2)}{1 + K/s(s+ 1)(s + 2)}$$[/tex]
On simplifying, we get: [tex]$$\frac{C(s)}{R(s)} = \frac{K}{s^3 + 3s^2 + 2s + K}$$[/tex]
The characteristic equation of the closed-loop system is: [tex]$$s^3 + 3s^2 + 2s + K = 0$$[/tex]
To obtain a range of values of K for stability, we will apply Routh-Hurwitz criterion. For that we need to form Routh array using the coefficients of s³, s², s and constant in the characteristic equation: $$\begin{array}{|c|c|} \hline s^3 & 1\quad 2 \\ s^2 & 3\quad K \\ s^1 & \frac{6-K}{3} \\ s^0 & K \\ \hline \end{array}$$
For stability, all the coefficients in the first column of the Routh array must be positive: [tex]$$1 > 0$$$$3 > 0$$$$\frac{6-K}{3} > 0$$[/tex]
Hence, [tex]$\frac{6-K}{3} > 0$[/tex] which implies $K < 6$.
So, the range of K for stability of the given control system is $0 < K < 6$.Therefore, the answer is : Range of K for stability of a unity feedback control system whose open-loop transfer function is K G(s) = K/s(s+ 1)(s + 2) is 0 < K < 6.
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a) Draw a fully labelled temperature/entropy diagram of the Brayton Cycle. b) Using appropriate thermodynamic terms, explain the Brayton cycle.
The Brayton cycle is a thermodynamic cycle that uses constant pressure in its heat rejection and heat addition processes. It is a cycle that operates in open systems.
Explanation of the Brayton cycle using appropriate thermodynamic terms:
The Brayton cycle is a thermodynamic cycle that uses gas turbines to generate power. It is a cycle that consists of four main processes: , heating, expansion, and cooling. The thermodynamic terms that are relevant to the Brayton cycle are the First Law of Thermodynamics, Second Law of Thermodynamics, and the Ideal Gas Law. The First Law of Thermodynamics states that energy cannot be created or destroyed but can only be transferred from one form to another. In the Brayton cycle, energy is converted from mechanical energy into thermal energy and then back into mechanical energy.
The Second Law of Thermodynamics states that all systems tend to move towards a state of maximum entropy. The Brayton cycle aims to minimize entropy and maximize efficiency. The Ideal Gas Law is a law that describes the behavior of ideal gases. In the Brayton cycle, the Ideal Gas Law is used to describe the behavior of the gas as it passes through the compressor, combustion chamber, turbine, and heat exchanger.
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A 14.568 kg uniform solid disk of radius 0.59 mis pin supported at its center. It is acted upon by a constant couple moment M 9.088 N.m. Starting from rest, determine the angular velocity in rad seconds when time is 2 seconds. The answer should be positive)
A 14.568 kg uniform solid disk of radius 0.59 mis pin supported at its center. It is acted upon by a constant couple moment M 9.088 N.m. Starting from rest, determine the angular velocity in rad seconds when time is 2 seconds.
In the given problem, we have to determine the angular velocity of a uniform solid disk of radius 0.59 m, mass 14.568 kg and acted upon by a constant couple moment M 9.088 N.m when time is 2 seconds.So,The moment of inertia of a uniform solid disk about its center is given as;I=12MR2Substitute the values of M and R in the above equation we get;I = 12(14.568 kg)(0.59 m)2= 2.554 kg m²The torque provided by the couple moment, τ is given as;τ = IαWhere α is the angular acceleration, thenα = τ/ISubstitute the values of τ and I in the above equation we get;α = (9.088 N.m) / (2.554 kg m²)= 3.562 rad/s²The final angular velocity is given by the formula;ωf = ωi + αtωi is the initial angular velocity, which is zero as the disc starts from rest. Substitute the values of α and t in the above equation we get;ωf = 0 + (3.562 rad/s²)(2 s) = 7.124 rad/s Therefore, the angular velocity when time is 2 seconds is 7.124 rad/s.
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For the system shown, the strain energy under load P is P2L3/2(kL3+3EI) For EI=30kN⋅m2,k=15kN/m,L=1 m, and P=900 N, the deflection under P is best given by a. 6.21 mm b. 5.00 mm c. 7.20 mm d. 8.57 mm
The deflection under the load P is 8.57 mm. Therefore, the correct answer is option D.
Given that, EI=30 kN.m², k = 15 kN/m, L=1 m, and P=900 N
The strain energy under the load of 900 N is given by:
U = (900 N)²×(1 m)³/(2 × (15 kN/m×(1 m)³+3×30 kN.m²))
= 8100/(540+90)
= 8100/630
= 12.7 J
The deflection under the load is given by:
δ = (P×L³)/(3×EI)
= (900 N×(1 m)³)/(3×30 kN.m²)
= 8.57 mm
Therefore, the correct answer is option D.
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"Your question is incomplete, probably the complete question/missing part is:"
For the system shown, the strain energy under load P is p²L³/2(kL³+3EI).
For EI=30 kN.m², k = 15 kN/m, L=1 m, and P=900 N, the deflection under P is best given by
a) 6.21 mm
b) 5.00 mm
c) 7.20 mm
d) 8.57 mm
pV.A (where p denotes pressure, V denotes flov velocity, and A is the cross-sectional area) indicates a Flow Work b Enthalpy c Shaft Work d Internal Energy
The formula pV.A is a representation of flow work. It is a significant term in thermodynamics that indicates the work done by fluids while flowing. Flow work, also known as flow energy or work of flow, refers to the work done by the fluid as it flows through the cross-sectional area of the pipeline in which it is flowing.
Flow work is an essential component of thermodynamics because it is the work required to move a fluid element from one point to another. It is dependent on both the pressure and volume of the fluid. A fluid's flow work can be calculated by multiplying the pressure by the volume and the cross-sectional area through which the fluid flows. As a result, the formula pV.A is a representation of flow work.
The formula pV.A does not indicate enthalpy, shaft work, or internal energy. Enthalpy, also known as heat content, is a measure of the energy required to transform a system from one state to another. Shaft work, on the other hand, refers to the work done by a mechanical shaft to move an object.
Internal energy, refers to the total energy of a system. flow work is the term indicated by the formula pV.A.
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Padding with zeros Example: Consider a four point sequence x(n)={1, 2, 3, 4). Find its a) 4-point DFT. b) 5-point DFT c) 1000-point DFT.
The answer to this part will be the same as the answer to part (b) since padding zeros does not affect the frequency content of the sequence, only its length, the 1000-point DFT is: X(0)=10, X(1)=-2-6i, X(2)=0, X(3)=2+6i, and X(4)=-2+2i.
When you are asked to pad zeros to a point sequence, you are expected to add zeros at the end of the point sequence to match a certain length. For example, in a four-point sequence x(n)={1, 2, 3, 4}, padding zeros to the sequence would involve adding zeros to the end of the sequence to meet a specified length, e.g., if the length required is 5 points, then zeros will be padded to the end of the sequence to get {1, 2, 3, 4, 0}.To solve the problem, we would use the following formula for computing DFT:X(k) = Summation [n=0, N-1] {x(n) exp(-i(2π/N)nk)}
Therefore, the 4-point DFT is: X(0)=10, X(1)=-2-6i, X(2)=0, and X(3)=2+6ib) 5-point DFT:To obtain the 5-point DFT of the sequence x(n)={1, 2, 3, 4}, we have to pad zeros to the end of the sequence such that the sequence has 5 points, i.e., x(n)={1, 2, 3, 4, 0}.Using the formula above and substituting the values for x(n), we get: X(k) = x(0) + x(1)exp(-i(2π/N)nk) + x(2)exp(-i(2π/N)2nk) + x(3)exp(-i(2π/N)3nk) + x(4)exp(-i(2π/N)4nk)Substituting x(n) = {1, 2, 3, 4, 0} into the above equation yields:X(0) = 1 + 2 + 3 + 4 + 0 = 10X(1) = 1 + 2exp(-iπ/2) + 3exp(-iπ) + 4exp(-i3π/2) + 0 = 1 - 2i - 3 - 4i = -2 - 6iX(2) = 1 + 2exp(-iπ) + 3exp(-i2π) + 4exp(-i3π) + 0 = 1 - 2 - 3 + 4 = 0X(3) = 1 + 2exp(-i3π/2) + 3exp(-i3π) + 4exp(-i9π/2) + 0 = 1 + 2i - 3 + 4i = 2 + 6iX(4) = 1 + 2exp(-i4π/2) + 3exp(-i4π) + 4exp(-i6π) + 0 = 1 + 2i - 3 - 4i = -2 + 2iTherefore, the 5-point DFT is: X(0)=10, X(1)=-2-6i, X(2)=0, X(3)=2+6i, and X(4)=-2+2ic) 1000-point DFT:
To obtain the 1000-point DFT of the sequence x(n)={1, 2, 3, 4}, we have to pad zeros to the end of the sequence such that the sequence has 1000 points, i.e., x(n)={1, 2, 3, 4, 0, 0, 0, ...}.Using the formula above and substituting the values for x(n), we get: X(k) = x(0) + x(1)exp(-i(2π/N)nk) + x(2)exp(-i(2π/N)2nk) + x(3)exp(-i(2π/N)3nk) + ... + x(999)exp(-i(2π/N)999nk)Since N=1000, the above formula will involve computing 1000 terms. For a large number like this, it is easier to compute using an algorithm known as the Fast Fourier Transform (FFT) instead of manually computing each term.
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Consider a Y-connected AC generator with a number of turns per phase of 600 turns. Find the flux per pole needed to produce the RMS generated line voltage of 4500 Volts at a frequency f-60 Hz. Select one: O a. Flux per pole = 28.2 mWebers O b. Flux per pole = 16.2 mWebers O c. None O d. Flux per pole = 19.85 mWebers O e. Flux per pole = 22.9 mWebers
Given, number of turns per phase, N = 600, RMS generated line voltage, V = 4500 V and frequency, f = 60 Hz. The relationship between RMS generated line voltage, V, frequency, f, and flux per pole, φ is given by the formula,V = 4.44fNφSo, the expression for flux per pole, φ is given by,φ = V / 4.44fNPlugging the given values, we get,φ = 4500 / (4.44 × 60 × 600)φ = 19.85 mWebers Therefore,
the flux per pole needed to produce the RMS generated line voltage of 4500 Volts at a frequency f-60 Hz is 19.85 mWebers.Option (D) is correct.Note: In AC generators, the voltage generated is proportional to the flux per pole, number of turns per phase, and frequency. The above formula is known as the EMF equation of an alternator.
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The hydraulic cylinder FC extends with a constant speed of 2 m/s and in turn rotates at point F. For the position shown, determine the angular acceleration of the cylinder and the acceleration of the box at point G (length FC 1000 mm).
The angular acceleration of the hydraulic cylinder is zero, and the acceleration of the box at point G is 2 m/s².
The given information states that the hydraulic cylinder FC extends with a constant speed of 2 m/s. Since the speed is constant, it implies that the cylinder is moving with a constant velocity, which means there is no acceleration in the linear motion of the cylinder.
Therefore, the angular acceleration of the cylinder is zero.As for the box at point G, its acceleration can be determined by analyzing the motion of the cylinder.
Since the cylinder rotates at point F, the box at point G will experience a centripetal acceleration due to its radial distance from the axis of rotation. This centripetal acceleration can be calculated using the formula:
Acceleration (a) = Radius (r) × Angular Velocity (ω)²
In this case, the radius is given as the length FC, which is 1000 mm (or 1 meter). Since the angular velocity is not provided, we can determine it by dividing the linear velocity of the cylinder by the radius of rotation.
Given that the linear velocity is 2 m/s and the radius is 1 meter, the angular velocity is 2 rad/s.
Substituting these values into the formula, we get:
Acceleration (a) = 1 meter × (2 rad/s)² = 4 m/s²
Hence, the acceleration of the box at point G is 4 m/s².
The angular acceleration of the hydraulic cylinder is zero because it is moving with a constant velocity. This means that there is no change in its rotational speed over time.
The acceleration of the box at point G is determined by the centripetal acceleration caused by the rotational motion of the cylinder. The centripetal acceleration depends on the radial distance from the axis of rotation and the angular velocity.
By calculating the radius and determining the angular velocity, we can find the centripetal acceleration. In this case, the centripetal acceleration of the box at point G is 4 m/s².
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The convolution expression in the time domain is transformed into multiplication in the s-domain as: L[x₁ (t) * x₂ (t)] = x₁(s).X₂ (s) Using x₁ (t) = u(t) - u(t-5) and x₂ (t) = u(t)- u(t-10), evaluate its convolution in time domain and then perform its equivalent in s-domain. Plot and compare the output in both domains.
To calculate the convolution of x₁(t) and x₂(t), let's apply the formula of convolution, which is denoted by -
[tex]x₁(t) * x₂(t).x₁(t) * x₂(t) = ∫ x₁(τ) x₂(t-τ) dτ= ∫ (u(τ) - u(τ-5))(u(t-τ) - u(t-τ-10)) dτIt[/tex]should be noted that u(τ-5) and u(t-τ-10) have a time delay of 5 and 10, respectively, which means that if we move τ to the right by 5,
After finding x₁(t) * x₂(t), the Laplace transform of the function is required. The Laplace transform is calculated using the formula:
L{x(t)} = ∫ x(t) * e^(-st) dt
L{(15-t)u(t)} = ∫ (15-t)u(t) * e^(-st) dt
= e^(-st) ∫ (15-t)u(t) dt
= e^(-st) [(15/s) - (1/s^2)]
L{(t-5)u(t-5)} = e^(-5s) L{t*u(t)}
= - L{d/ds(u(t))}
= - L{(1/s)}
= - (1/s)
L{(t-10)u(t-10)} = e^(-10s) L{t*u(t)}
= - L{d/ds(u(t))}
= - L{(1/s)}
= - (1/s)
L{(15-t)u(t) - (t-5)u(t-5) + (t-10)u(t-10)} = (15/s) - (1/s^2) + (1/s)[(1-e^(-5s))(t-5) + (1-e^(-10s))(t-10)]
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The minimum pressure on an object moving horizontally in water (Ttemperatu at10 degree centrigrade) at (x + 5) mm/s (where x is the last two digits of your student ID) at a depth of 1 m is 80 kPa (absolute). Calculate the velocity that will initiate cavitation. Assume the atmospheric pressure as 100 kPa (absolute). x = 98
The velocity that will initiate cavitation is approximately 2827.6 mm/s or 37.12 mm/s
To calculate the velocity that will initiate cavitation, we can use the Bernoulli's equation between two points along the flow path. The equation relates the pressure, velocity, and elevation at those two points.
In this case, we'll compare the conditions at the minimum pressure point (where cavitation occurs) and a reference point at the same depth.
The Bernoulli's equation can be written as:
[tex]\[P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2\][/tex]
where:
[tex]\(P_1\)[/tex] and [tex]\(P_2\)[/tex] are the pressures at points 1 and 2, respectively,
[tex]\(\rho\)[/tex] is the density of water,
[tex]\(v_1\)[/tex] and [tex]\(v_2\)[/tex] are the velocities at points 1 and 2, respectively,
[tex]\(g\)[/tex] is the acceleration due to gravity, and
[tex]\(h_1\)[/tex] and [tex]\(h_2\)[/tex] are the elevations at points 1 and 2, respectively.
In this case, we'll consider the minimum pressure point as point 1 and the reference point at the same depth as point 2.
The elevation difference between the two points is zero [tex](\(h_1 - h_2 = 0\))[/tex]. Rearranging the equation, we have:
[tex]\[P_1 - P_2 = \frac{1}{2} \rho v_2^2 - \frac{1}{2} \rho v_1^2\][/tex]
Given:
[tex]\(P_1 = 80 \, \text{kPa}\)[/tex] (absolute pressure at the minimum pressure point),
[tex]\(P_2 = 100 \, \text{kPa}\)[/tex] (atmospheric pressure),
[tex]\(\rho\) (density of water at 10 °C)[/tex] can be obtained from a water density table as [tex]\(999.7 \, \text{kg/m}^3\)[/tex], and
[tex]\(v_1 = (98 + 5) \, \text{mm/s} = 103 \, \text{mm/s}\).[/tex]
Substituting the values into the equation, we can solve for [tex]\(v_2\)[/tex] (the velocity at the reference point):
[tex]\[80 \, \text{kPa} - 100 \, \text{kPa} = \frac{1}{2} \cdot 999.7 \, \text{kg/m}^3 \cdot v_2^2 - \frac{1}{2} \cdot 999.7 \, \text{kg/m}^3 \cdot (103 \, \text{mm/s})^2\][/tex]
Simplifying and converting the units:
[tex]\[ -20 \, \text{kPa} = 4.9985 \, \text{N/m}^2 \cdot v_2^2 - 0.009196 \, \text{N/m}^2 \cdot \text{m}^2/\text{s}^2\][/tex]
Rearranging the equation and solving for \(v_2\):
[tex]\[v_2^2 = \frac{-20 \, \text{kPa} + 0.009196 \, \text{N/m}^2 \cdot \text{m}^2/\text{s}^2}{4.9985 \, \text{N/m}^2} \]\\\\\v_2^2 = 7.9926 \, \text{m}^2/\text{s}^2\][/tex]
Taking the square root to find [tex]\(v_2\)[/tex]:
[tex]\[v_2 = \sqrt{7.9926} \, \text{m/s} \approx 2.8276 \, \text{m/s}\][/tex]
Converting the velocity to millimeters per second:
[tex]\[v = 2.8276 \, \text{m/s} \cdot 1000 \, \text{mm/m} \approx 2827.6 \, \text{mm/s}\][/tex]
Therefore, the velocity that will initiate cavitation is approximately 2827.6 mm/s or 37.12 mm/s (rounded to two decimal places).
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The output of a linear variable differential transformer is connected to a 5 V Voltmeter through an amplifier whose amplification factor is 250. An output of two mV appears across the terminals off LVDT when the core moves through a distance of 0.5 mm. Calculate the sensitivity of the LVDT and that of the whole setup. The milli-voltmeter scale has 100 divisions. the scale can be read to 1/5 of a division. Calculate the resolution of the instrument in mm. [E 5.3]
Therefore, the resolution of the instrument is 2 mm.
The LVDT (Linear Variable Differential Transformer) is a type of transducer that produces an output voltage that varies linearly with the displacement of the core. This type of transducer has applications in the measurement of position, acceleration, vibration, and other physical parameters.
Let's solve the given problem step by step:
Sensitivity of the LVDT:
Sensitivity of the LVDT is defined as the ratio of the output voltage to the input displacement.
Mathematically, it is given by the following formula:
Sensitivity of LVDT = Output voltage/ Displacement of core
Given that, an output of 2 mV appears across the terminals of LVDT when the core moves through a distance of 0.5 mm.
Therefore, the sensitivity of the LVDT is:
Sensitivity of LVDT = Output voltage/ Displacement of core= (2 mV/0.5 mm) = 4 mV/mm
Sensitivity of the whole setup:
Sensitivity of the whole setup is defined as the ratio of the output voltage of the system to the input physical parameter being measured (displacement in this case).Mathematically, it is given by the following formula:
Sensitivity of the whole setup = (Output voltage of the system/ Input physical parameter) x Amplification factor
Given that, the output of the LVDT is connected to a 5 V voltmeter through an amplifier whose amplification factor is 250.
Therefore, the sensitivity of the whole setup is:
Sensitivity of the whole setup = (Output voltage of the system/ Input physical parameter) x Amplification factor= (2 mV/0.5 mm) x 250 = 1000 mV/mm
Resolution of the instrument:
Resolution of the instrument is the smallest increment that can be detected on the scale of the instrument. In this case, the voltmeter scale has 100 divisions, and it can be read to 1/5 of a division.
Therefore, the smallest increment that can be detected on the scale is:
Smallest increment = (1/5) x (1/100) = 0.002 V
To find the resolution of the instrument in mm, we need to convert the voltage reading into displacement reading using the sensitivity of the whole setup.
Resolution of the instrument = Smallest increment x Sensitivity of the whole setup= 0.002 V x 1000 mV/mm= 2 mm
Therefore, the resolution of the instrument is 2 mm.
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(b) Moist air enters a duct at 10∘C,80%RH, and a volumetric flow rate of 150 m³/min. The mixture is heated as it flows through the duct and exits at 30∘C. No moisture is added or removed, and the mixture pressure remains approximately constant at 1 bar. For steady-state operation; i. sketch on T−s diagram the heating process, and determine; ii. the rate of heat transfer, in kJ/min; and iii. the relative humidity at the exit.
The problem involves moist air entering a duct at specific conditions and being heated as it flows through. The goal is to determine the heating process on a T-s diagram, calculate the rate of heat transfer, and find the relative humidity at the exit.
ii. To determine the rate of heat transfer, we can use the energy balance equation for the process. The rate of heat transfer can be calculated using the equation Q = m_dot * (h_exit - h_inlet), where Q is the heat transfer rate, m_dot is the mass flow rate of the moist air, and h_exit and h_inlet are the specific enthalpies at the exit and inlet conditions, respectively.
iii. The relative humidity at the exit can be determined by calculating the saturation vapor pressure at the exit temperature and dividing it by the saturation vapor pressure at the same temperature. This can be expressed as RH_exit = (P_vapor_exit / P_sat_exit) * 100%, where P_vapor_exit is the partial pressure of water vapor at the exit and P_sat_exit is the saturation vapor pressure at the exit temperature.
In order to sketch the heating process on a T-s diagram, we need to determine the specific enthalpy and entropy values at the inlet and exit conditions. With these values, we can plot the process line on the T-s diagram. By solving the equations and performing the necessary calculations, the rate of heat transfer and the relative humidity at the exit can be determined, providing a complete analysis of the problem.
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A. Also find the minimum cable gradient at C. (c) Taking the tension in the cable at A as 1000kN and the cable slope as 15 ∘, determine the reaction at A if the cable passes over (i) a saddle, (ii) a pulley there to form a backstay at 30 ∘to the horizontal. [835,2kN;800,0kN;896,5kN;857,4kN,1,52 ∘. (i) 816,5kN (ii) 765kN;7,5∘]
(i) When the cable passes over a saddle, the reaction at A is 866.03 kN. (ii) When the cable passes over a pulley to form a backstay at 30 degree to the horizontal, the reaction at A is 861.03 kN in the downward direction.
(i) When the cable passes over a saddle, the angle of the cable is not specified. So, let's assume that the angle of the cable is 30 degrees.
The horizontal component of tension can be found using the formula:
Horizontal component of tension = Tension x cos(theta)
Where Tension is the tension in the cable and theta is the angle of the cable with respect to the horizontal.
Horizontal component of tension = 1000kN x cos(30)
= 866.03 kN
Since the horizontal component of tension at A is equal to the reaction at A, the reaction at A is 866.03 kN.
(ii) When the cable passes over a pulley to form a backstay at 30 ∘to the horizontal, the angle of the cable is specified.
The vertical component of tension can be found using the formula:
Vertical component of tension = Tension x sin(theta)
Where Tension is the tension in the cable and theta is the angle of the cable with respect to the horizontal.
Vertical component of tension = 1000kN x sin(30) = 500 kN
The component of tension perpendicular to the backstay can be found using the formula:
Component of tension perpendicular to backstay = Tension x sin(60)
Where Tension is the tension in the cable and 60 degrees is the complement of the 30 degrees angle between the cable and backstay.
Component of tension perpendicular to backstay = 1000kN x sin(60)
= 866.03 kN
The weight of the object is not given in the problem statement, so let's assume it to be 5000 N.
Using the formula for the vertical component of tension and the component of tension perpendicular to the backstay, we can find the vertical component of the reaction at A:
Vertical component of reaction at A = Weight of object - Component of tension perpendicular to backstay - Vertical component of tension
Vertical component of reaction at A = 5000 N - 866.03 kN - 500 kN
= - 861.03 kN
The negative sign indicates that the reaction at A is acting in the downward direction.
Therefore, the reaction at A is 861.03 kN in the downward direction.
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The complete question is attached below:
Engine oil is to be cooled from 80 to 50 oC by using counter flow, concentric tube heat
exchanger with cooling water available at 20oC. Water flows inside a tube with an ID of Di = 2.5 cm at a
rate of 0.08 kg/s and oil flows through the annulus at a rate of 0.016 kg/s. The heat transfer coefficient
for the water side and oil side are respectively, 1000 W/m2 oC and 80 W/m2 oC ; the fouling factor are
Fwater=0.00018 oC/W and Foil=0.00018 oC/W; and the tube wall resistance is negligible. Calculate the tube
length required.
To cool engine oil from 80 to 50°C using a counterflow, concentric tube heat exchanger, with cooling water available at 20°C, the required tube length needs to be calculated. The problem provides information on flow rates, heat transfer coefficients.
To determine the tube length required, we can use the basic equation for heat transfer in a heat exchanger:
Q = U × A × ΔTlm
where Q is the heat transfer rate, U is the overall heat transfer coefficient, A is the heat transfer area, and ΔTlm is the logarithmic mean temperature difference.
In this case, the heat transfer rate can be calculated as the product of the mass flow rate and specific heat capacity difference of the oil:
Q = m_oil × Cp_oil × ΔT_oil
The overall heat transfer coefficient can be calculated using the individual heat transfer coefficients for the water and oil sides, as well as the fouling factors:
1/U = (1/h_water) + (1/h_oil) + (Rf_water) + (Rf_oil)
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This section is for postgraduate (Masters) students only. While undergraduate (bachelors) students are free to attempt this part of the practical it will not contribute to the report grade. When using the inter event interval to measure the rotational speed of a motor the rate at which information is available to the system is proportional to the rotational speed of the motor. This means that a control system need not be limited to a fixed interval to calculate parameters as has been the case in the previous parts of this practical. 1) If the loop rate of the system were increased by a factor of 2 what would happen to the P, I and D gains in the microcontroller code if you wanted to keep the same system tuning? [3 marks] 2) In practice would you expect the noise from the derivative component of the control system to increase, decrease or stay the same with an increasing loop rate? Why? [2 marks] 3) Detail the steps required to produce a variable rate control system on the Flinduino. You are not required to actually produce such a system, just explain how it could be done. [20 marks] Note: since part 13) is a design question the marking criteria does not follow the standard 1 mark for 1 point/sentence/calculation that the rest of the practical does. You are not expected to write a large amount, just come up with a solid outline of how the task could be achieved. Dot-points are acceptable.
1) If the loop rate of the system were increased by a factor of 2, then the P, I, and D gains in the microcontroller code required to keep the same system tuning would have to be halved.2) In practice, the noise from the derivative component of the control system would increase as the loop rate increases.
This is because the derivative component of the control system amplifies high-frequency noise, which is why higher loop rates can amplify noise even further.3) To produce a variable rate control system on the Flinduino, the following steps are required:Identify the motor and control system's required transfer function based on the given input and output parameters.
Test and refine the control system by changing the input parameters and monitoring the output to ensure the motor's stability and response time is optimal.This variable rate control system allows the system to maintain optimal control of the motor's speed under changing load conditions, and is particularly useful for applications that require precise control of the motor's speed.
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Routh-Hurwitz stability criterion Given the unity feedback system: G(s)=(s 6+2s5+3s4+4s3+5s26s−7)8
Using the code. Modify and correct the given code so that it will solve the following - Routh Table - Stability of the system - Number of poles on the right hand side of the (s) plane - Poles of the system
% Code By
% Farzad Sagharchi ,Iran
% 2007/11/12
coeffVector = input('input vector of your system coefficients: \n i.e. [an an-1 an-2 ... a0] = ');
ceoffLength = length(coeffVector);
rhTableColumn = round(ceoffLength/2);
rhTable = zeros(ceoffLength,rhTableColumn);
rhTable(1,:) = coeffVector(1,1:2:ceoffLength);
if (rem(ceoffLength,2) ~= 0)
rhTable(2,1:rhTableColumn - 1) = coeffVector(1,2:2:ceoffLength);
else
rhTable(2,:) = coeffVector(1,2:2:ceoffLength);
end
epss = 0.01;
for i = 3:ceoffLength
if rhTable(i-1,:) == 0
order = (ceoffLength - i);
cnt1 = 0;
cnt2 = 1;
for j = 1:rhTableColumn - 1
rhTable(i-1,j) = (order - cnt1) * rhTable(i-2,cnt2);
cnt2 = cnt2 + 1;
cnt1 = cnt1 + 2;
end
end
for j = 1:rhTableColumn - 1
firstElemUpperRow = rhTable(i-1,1);
rhTable(i,j) = ((rhTable(i-1,1) * rhTable(i-2,j+1)) - ....
(rhTable(i-2,1) * rhTable(i-1,j+1))) / firstElemUpperRow;
end
if rhTable(i,1) == 0
rhTable(i,1) = epss;
end
end
unstablePoles = 0;
for i = 1:ceoffLength - 1
if sign(rhTable(i,1)) * sign(rhTable(i+1,1)) == -1
unstablePoles = unstablePoles + 1;
end
end
fprintf('\n Routh-Hurwitz Table:\n')
rhTable
if unstablePoles == 0
fprintf('~~~~~> it is a stable system! <~~~~~\n')
else
fprintf('~~~~~> it is an unstable system! <~~~~~\n')
end
fprintf('\n Number of right hand side poles =%2.0f\n',unstablePoles)
reply = input('Do you want roots of system be shown? Y/N ', 's');
if reply == 'y' || reply == 'Y'
sysRoots = roots(coeffVector);
fprintf('\n Given polynomial coefficients roots :\n')
sysRoots
end
The correct code for solving Routh Table - Stability of the system - Number of poles is coded below.
The corrected and modified code to solve the Routh-Hurwitz stability criterion:
coeffVector = input('Input vector of your system coefficients: \n i.e. [an an-1 an-2 ... a0] = ');
coeffLength = length(coeffVector);
rhTableColumn = ceil(coeffLength/2);
rhTable = zeros(coeffLength, rhTableColumn);
rhTable(1, :) = coeffVector(1, 1:2:coeffLength);
if (rem(coeffLength, 2) ~= 0)
rhTable(2, 1:rhTableColumn - 1) = coeffVector(1, 2:2:coeffLength);
else
rhTable(2, :) = coeffVector(1, 2:2:coeffLength);
end
epss = 0.01;
for i = 3:coeffLength
if all(rhTable(i-1, :) == 0)
order = (coeffLength - i);
cnt1 = 0;
cnt2 = 1;
for j = 1:rhTableColumn - 1
rhTable(i-1, j) = (order - cnt1) * rhTable(i-2, cnt2);
cnt2 = cnt2 + 1;
cnt1 = cnt1 + 2;
end
end
for j = 1:rhTableColumn - 1
firstElemUpperRow = rhTable(i-1, 1);
rhTable(i, j) = ((rhTable(i-1, 1) * rhTable(i-2, j+1)) - ...
(rhTable(i-2, 1) * rhTable(i-1, j+1))) / firstElemUpperRow;
end
if rhTable(i, 1) == 0
rhTable(i, 1) = epss;
end
end
unstablePoles = 0;
for i = 1:coeffLength - 1
if sign(rhTable(i, 1)) * sign(rhTable(i+1, 1)) == -1
unstablePoles = unstablePoles + 1;
end
end
fprintf('\nRouth-Hurwitz Table:\n')
rhTable
if unstablePoles == 0
fprintf('~~~~~> It is a stable system! <~~~~~\n')
else
fprintf('~~~~~> It is an unstable system! <~~~~~\n')
end
fprintf('\nNumber of right-hand side poles: %d\n', unstablePoles)
reply = input('Do you want the roots of the system to be shown? Y/N ', 's');
if reply == 'y' || reply == 'Y'
sysRoots = roots(coeffVector);
fprintf('\nGiven polynomial coefficients roots:\n')
sysRoots
end
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An AISI 1018 steel has a yield strenght, Sy = 295 MPa. Using the distortion-energy theory for the following given state of plane stress, determine the factor of safety. Write your final answer in two decimal places. σx =−89MPa, σy = 40MPa, τxy = 0 Hints: For distortion energy theory: σ′ = (σₓ² − σₓσᵧ + σᵧ² + 3rₓᵧ²)¹/²
n = Sy/σ'
To determine the factor of safety using the distortion-energy theory, we need to calculate σ' and then find the factor of safety (n) using the formula n = Sy/σ'.
Given:
σx = -89 MPa
σy = 40 MPa
τxy = 0
First, we need to calculate σ':
σ' = (√(σx² - σxσy + σy² + 3τxy²))
Substituting the given values:
σ' = (√((-89)² - (-89)(40) + (40)² + 3(0)²))
σ' = (√(7921 + 3560 + 1600 + 0))
σ' = (√13081)
σ' ≈ 114.41 MPa
Now, we can calculate the factor of safety (n):
n = Sy/σ'
n = 295 MPa / 114.41 MPa
n ≈ 2.58
Therefore, the factor of safety is approximately 2.58.
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Given the following C program: int main()
{ int index; double data[3]; GetData(\&data[0], \&data[1], \&data[2]);
printf("Index Dataln")
; for (index =0; index <3; indext++) {
printf("\%sd %8.31f(n ", index, data[index]); }
getch(); return θ; } The main function creates a double array with 3 elements and then passes all three elements (individually) to the GetData function. The main then prints the three values from the data array along with their element numbers. Complete the program by creating the function GetData that works as follows: 1. The function must assign the value 7.5 to element 0 of the data array. 2. The function must ask the user what value to assign to element 1 of the data array and input that value from the user. Make sure that you use the pointer representing the array element directly in your scanf (that is, you cannot input into a simple variable and then assign to the element). 3. The function must add the value from element 0 and element 1 of the array and assign the sum to element 2 of the data array (make sure you are retrieving the value from element 0 and not just hardcoding the 7.5). An execution of the program might look as follows: What value would you like to assign to element 1?10,5
Index Data
0 7.500
1 10.500
2 18.000
In the main function, a double array of 3 elements is created and passed individually to the GetData function. The main function then prints the three values from the data array along with their element numbers.
#include
#include void GetData(double *ptr1, double *ptr2, double *ptr3)
{ *ptr1 = 7.5; printf("\nWhat value would you like to assign to element 1?");
scanf("%lf", ptr2); *ptr3 = (*ptr1) + (*ptr2); return; }
int main() { int index; double data[3];
GetData(&data[0], &data[1], &data[2]);
printf("Index Data\n");
for (index = 0; index < 3; index++) { printf("%d %8.3lf\n", index, data[index]); }
getch();
return 0; }
The value 7.5 is assigned to element 0 of the data array.2. The user is asked what value to assign to element 1 of the data array, and that value is inputted from the user. The pointer representing the array element is used directly in the scanf.3. The value from element 0 and element 1 of the array is added, and the sum is assigned to element 2 of the data array.
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An ammonia (R717) heat pump is used to heat hot water for CIP (cleaning) fluid, for a food manufacturing facility. The heat pump will heat water from 50°C to 90° and provide 1 MW of heating. The heat pump will operate with an evaporation temperature of 10°C and a condensing temperature of 100°C. It is proposed to use the evaporator of the heat pump to keep the air in a processing room climate controlled at 15°C. Chilling is needed to maintain the air temperature, as there is considerable heating of the air due to processing equipment operating in the room. What is the amount of chilling at 15°C that can be provided by the heat pump? (kW)
Given data:Heat pump will provide 1 MW of heating.The heat pump will operate with an evaporation temperature of 10°C and a condensing temperature of 100°C.The evaporator of the heat pump is used to keep the air in a processing room climate controlled at 15°C.
The heat pump provides heating of water from 50°C to 90°C.To find: The amount of chilling at 15°C that can be provided by the heat pumpSolution:As per the question, the evaporator of the heat pump is used to keep the air in a processing room climate controlled at 15°C.Evaporation temperature of the heat pump is 10°C, so the heat is extracted at 10°C from the room.
The heat extracted by the evaporator of the heat pump, as refrigeration,Q = 1 / COP * W = (m * c * ΔT) / COPWe have to calculate W, soW = m * c * ΔT * COPW = 1.225 * V * 0.718 * (-10) * 3W = - 26.23 VAt 15°C, the volume of the room would be known so we can easily calculate W as per the above equation.So, the amount of chilling at 15°C that can be provided by the heat pump is -26.23 V (kW).Negative sign indicates that the heat pump is absorbing heat from the room. Hence, the heat pump will act as a refrigerator in this case.
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A unity negative feedback control system has the loop transfer suction.
L(S)=G₁ (S) G (S) = K (S+2) / (S+1) (S+2.5) (S+4) (S+10) a) sketch the root lows as K varies from 0 to 2000 b) Find the roofs for K equal to 400, 500 and 600
A unity negative feedback control system has the loop transfer suction L(S) = G1(S)G(S) = K(S + 2) / (S + 1)(S + 2.5)(S + 4)(S + 10).a) Sketch the root lows as K varies from 0 to 2000:b) .
Find the roofs for K equal to 400, 500 and 600a) Root Locus is the plot of the closed-loop poles of the system that change as the gain of the feedback increases from zero to infinity. The main purpose of the root locus is to show the locations of the closed-loop poles as the system gain K is varied from zero to infinity.
The poles of the closed-loop transfer function T(s) = Y(s) / R(s) can be located by solving the characteristic equation. Therefore, the equation is given as:K(S+2) / (S+1)(S+2.5)(S+4)(S+10) = 1or K(S+2) = (S+1)(S+2.5)(S+4)(S+10)or K = (S+1)(S+2.5)(S+4)(S+10) / (S+2)Here, we can find out the closed-loop transfer function T(s) as follows:T(S) = K / [1 + KG(S)] = K(S+2) / (S+1)(S+2.5)(S+4)(S+10) .
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A unity negative feedback system has the loop transfer function L(s) = Gc (s)G(s) = (1 + p) s -p/s² + 4s + 10 Develop an m-file to obtain the root locus as p varies; 0 < p <[infinity]. For what values of p is the closed-loop stable?
The closed-loop system is stable for values of p between 0 and 10/3.
A unity negative feedback system has the loop transfer function L(s) = Gc(s)G(s)
= (1 + p)s - p/s² + 4s + 10.
In order to obtain the root locus as p varies, we need to write the open-loop transfer function as G(s)H(s)
= 1/L(s) = s² + 4s + 10/p - (1 + p)/p.
To obtain the root locus, we first need to find the poles of G(s)H(s).
These poles are given by the roots of the characteristic equation 1 + L(s) = 0.
In other words, we need to find the values of s for which L(s) = -1.
This leads to the equation (1 + p)s - p = -s² - 4s - 10/p.
Expanding this equation and simplifying, we get the quadratic equation s² + (4 - 1/p)s + (10/p - p) = 0.
Using the Routh-Hurwitz stability criterion, we can determine the values of p for which the closed-loop system is stable. The Routh-Hurwitz stability criterion states that a necessary and sufficient condition for the stability of a polynomial is that all the coefficients of its Routh array are positive.
For our quadratic equation, the Routh array is given by 1 10/p 4-1/p which means that the system is stable for 0 < p < 10/3.
The MATLAB code to obtain the root locus is as follows: num = [1 (4 - 1/p) (10/p - p)]; den = [1 4 10/p - (1 + p)/p]; rlocus (num, den, 0:0.1:100);
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Mention the following: a. Type of the materials used to make the windows and mention two advantages and disadvantages?
The types of materials commonly used to make windows include glass, vinyl, wood, and aluminum. Each material has its advantages and disadvantages.
Glass is a popular choice for windows due to its transparency, durability, and ability to let in natural light. It is also resistant to heat and moisture. However, glass windows can be fragile and may require additional measures for insulation.
Vinyl windows offer excellent energy efficiency, low maintenance, and affordability. They are resistant to moisture and do not require painting. However, they may not provide the same aesthetic appeal as other materials, and color options may be limited.
Wood windows offer a classic and natural look, enhancing the overall aesthetics of a space. They provide good insulation and can be customized with various finishes. However, wood requires regular maintenance, such as painting and sealing, to protect against moisture and rot.
Aluminum windows are known for their strength and durability. They are resistant to weathering, corrosion, and rot. Additionally, they offer a sleek and modern appearance. On the downside, aluminum windows are not as energy-efficient as other materials and may conduct heat and cold.
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Radiation question, please help and upvote will; be given for
support!
Briefly explain the difference between the specular reflection and diffuse reflection in the properties of radiation heat transfer. (5 marks)
The difference between the specular reflection and diffuse reflection in the properties of radiation heat transfer is that in specular reflection, the reflected wave is directional and is reflected at the same angle of incidence as it hits the surface, whereas in diffuse reflection, the reflected wave is not directional and is scattered in multiple directions.
Radiation heat transfer can be categorized into two types of reflections: specular reflection and diffuse reflection.
The properties of these two types of reflection differ from one another.
Specular Reflection is when an incident ray falls on a surface and bounces off at the same angle, preserving the angle of incidence and the angle of reflection.
The wave reflected in specular reflection is highly directional, that is, the surface is very smooth, and the angle of incidence is the same as the angle of reflection.
Diffuse reflection, on the other hand, is when an incident ray falls on a surface and bounces off in multiple directions.
This type of reflection is caused by rough surfaces that scatter the incoming wave. Unlike specular reflection, diffuse reflection is not directional.
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The flue gas (at atmospheric pressure) from a chemical plant contains hazardous vapors that must be condensed by lowering its temperature from 295°C to 32°C. The gas flow rate is 0.60 m ∧3/s. Water is available at 12°C at 1.5 kg/s. A counterflow heat exchanger will be used with water flowing through the tubes. The gas has a specific heat of 1.12 kJ/kg−K and a gas constant of 0.26 kJ/kg−K; let c pwater=4.186 kJ/kg−K. Calculate the logarithmic mean temperature difference (°C).(20pts) Draw and label the temperature-flow diagram. Round off your answer to three (3) decimal places.
The logarithmic mean temperature difference (LMTD) is 106.614°C.
The logarithmic mean temperature difference (LMTD) is used to compute the heat transfer rate in a heat exchanger or a cooling tower.
When a chemical plant's flue gas (at atmospheric pressure) contains harmful vapors that must be condensed by reducing its temperature from 295°C to 32°C and the gas flow rate is 0.60 m ∧3/s, this calculation becomes crucial. Water is available at 12°C at 1.5 kg/s.
A counterflow heat exchanger will be used with water flowing through the tubes.
The gas has a specific heat of 1[tex].12 kJ/kg−K[/tex]and a gas constant of 0.26 kJ/kg−K;
let c [tex]pwater=4.186 kJ/kg−K.[/tex]
The logarithmic mean temperature difference (LMTD) for the process is calculated as follows:
Step 1: Mean temperature of the hot fluid, [tex]ΔT1=(295−32)/ln(295/32)=175.364°C[/tex]
Step 2: Mean temperature of the cold fluid, [tex]ΔT2=(12−32)/ln(12/32)=20.609°C[/tex]
Step 3: Logarithmic mean temperature difference
[tex]ΔTlm= (ΔT1-ΔT2)/ ln(ΔT1/ΔT2) = (175.364 - 20.609)/ln(175.364/20.609) = 106.614°C.[/tex]
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(a) State the four (4) commonly implemented CFD discretization methods or programs. (b) Using a uiform geomtery and grid, sketch the discretization method for each of the methods in (a).
The four commonly implemented CFD discretization methods are - (FDM), (FVM), (FEM) and (SEM).
(a) The four commonly implemented CFD discretization methods or programs are as follows:
Finite difference method (FDM)
Finite volume method (FVM)
Finite element method (FEM)
Spectral element method (SEM)
(b) Sketch of discretization method for each of the methods in (a) using a uniform geometry and grid is as follows:
1. Finite difference method (FDM) In finite difference method, the discretization process divides the whole domain into a discrete grid or mesh, and the partial derivatives are replaced by difference equations.
2. Finite volume method (FVM)The finite volume method focuses on the conservation of mass, energy, and momentum. A control volume in which all the variables are considered to be constant is considered in the method.
3. Finite element method (FEM)In finite element method, the solution is approximated over a finite set of basis functions that are defined within each element of the mesh. The unknowns are determined using a variational principle, and the equation is then solved using a linear or nonlinear solver.
4. Spectral element method (SEM)The spectral element method combines the strengths of finite element and spectral methods. A spectral decomposition is performed within each element to obtain the solution, which is then used to interpolate the solution within the element. This method is highly accurate and efficient.
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Write a function M-file that implements (8) in the interval 0 ≤ t ≤ 55. Note that the initial condition must now be in the form [yo, v0, w0] and the matrix Y, output of ode45, has now three columns (from which y, v and w must be extracted). On the same figure, plot the three time series and, on a separate window, plot the phase plot using figure (2); plot3 (y,v,w); hold on; view ([-40,60]) xlabel('y'); ylabel('vay); zlabel('way''); Do not forget to modify the function defining the ODE. The output is shown in Figure 9. The limits in the vertical axis of the plot on the left were delib- erately set to the same ones as in Figure 8 for comparison purposes, using the MATLAB command ylim ([-2.1,2.1]). You can play around with the 3D phase plot, rotating it by clicking on the circular arrow button in the figure toolbar, but submit the plot with the view value view ([-40, 60]) (that is, azimuth = -40°, elevation = 60°).
The task at hand is to write a function M-file that implements (8) in the interval 0 ≤ t ≤ 55. The initial condition must now be in the form [yo, v0, w0]. The matrix Y, which is the output of ode45, now has three columns. Y(:,1) represents y, Y(:,2) represents v and Y(:,3) represents w. We need to extract these columns.
We also need to plot the three time series on the same figure and, on a separate window, plot the phase plot using figure (2); plot3 (y,v,w); hold on; view ([-40,60]) xlabel('y'); ylabel('vay); zlabel('way'').Here is a function M-file that does what we need:
function [tex]yp = fun(t,y)yp = zeros(3,1);yp(1) = y(2);yp(2) = y(3);yp(3) = -sin(y(1))-0.1*y(3)-0.1*y(2);[/tex]
endWe can now use ode45 to solve the ODE.
The limits in the vertical axis of the plot on the left were deliberately set to the same ones as in Figure 8 for comparison purposes, using the MATLAB command ylim ([-2.1,2.1]). You can play around with the 3D phase plot, rotating it by clicking on the circular arrow button in the figure toolbar, but submit the plot with the view value view ([-40, 60]) (that is, azimuth = -40°, elevation = 60°).
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A lift pump having a diameter of 4 inches and a stroke of 6 inches is used to lift water from a 20ft well and deliver it to a cylindrical tank at a height of 12ft. If the volumetric efficiency of the pump at 10 lifting strokes per minute is 90%, what is the pump capacity? Also compute for the power required to operate the pump manually if its mechanical efficiency is 80%? More over, How long does the pump is required to fully fill the 600 liter tank if its operating efficiency is 70%?
A lift pump is required to lift water from a well and deliver it to a cylindrical tank. The lift pump has a diameter of 4 inches and a stroke of 6 inches. The lift pump's volumetric efficiency at 10 lifting strokes per minute is 90%.The pump capacity can be calculated using the following formula:
Pump capacity = π × r² × s × n × VEF where r is the radius of the lift pump, s is the stroke of the lift pump, n is the number of lifting strokes per minute, and VEF is the volumetric efficiency factor.The diameter of the lift pump is given as 4 inches, which means that the radius is 2 inches.r = 2 inches = 0.167 feet
The stroke of the lift pump is given as 6 inches, which means that the stroke is 0.5 feet.s = 0.5 feet The number of lifting strokes per minute is given as 10.n = 10 The volumetric efficiency factor is given as 90%.VEF = 0.9Pump capacity = π × r² × s × n × VEF= 3.1416 × (0.167)² × 0.5 × 10 × 0.9= 0.746 cubic feet per minute (CFM)The power required to operate the pump manually can be calculated using the following formula:Power = F × s × n / 33000 where F is the force required to lift the water, s is the stroke of the lift pump, n is the number of lifting strokes per minute, and 33,000 is the conversion factor.The force required to lift the water can be calculated using the following formula:Force = Weight of water lifted / Mechanical efficiency where Mechanical efficiency is given as 80%.
Weight of water lifted = Density of water × Volume of water lifted
Density of water = 62.4 lb/ft³
Volume of water lifted = Pump capacity × Operating efficiency= 0.746 × 0.7= 0.522 cubic feet Weight of water lifted = 62.4 × 0.522 = 32.6288 lb Force = 32.6288 / 0.8 = 40.786 lb Power = F × s × n / 33000= 40.786 × 0.5 × 10 / 33000= 0.000619 horsepower (HP)
The lift pump has a capacity of 0.746 cubic feet per minute (CFM) and is required to fill a 600-liter cylindrical tank at a height of 12 feet. The operating efficiency of the lift pump is given as 70%.The time required to fully fill the 600-liter tank can be calculated using the following formula:Time = Volume of tank / Pump capacity / Operating efficiency where the volume of the tank is given as 600 liters.The volume of the tank needs to be converted from liters to cubic feet.1 liter = 0.0353147 cubic feet Therefore, 600 liters = 600 × 0.0353147 = 21.1888 cubic feet Time = 21.1888 / 0.746 / 0.7= 41.36 minutes
Therefore, the pump capacity is 0.746 cubic feet per minute (CFM).The power required to operate the pump manually is 0.000619 horsepower (HP).The pump is required to fully fill the 600-liter tank in 41.36 minutes.
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