The steady-state temperature distribution in a 3-D rectangular parallelepiped with the given boundary conditions, we can use the method of separation of variables.
We assume that the temperature distribution can be expressed as a product of three functions, each dependent on a single variable:
T(x, y, z) = X(x)Y(y)Z(z)
By substituting this into the heat equation and dividing both sides by the temperature T, we can separate the variables and obtain three ordinary differential equations:
[tex]X''(x)/X(x) = -(Y''(y)/Y(y) + Z''(z)/Z(z)) = -λ²[/tex]
Here, λ² is the separation constant.
We can solve each equation separately:
1) X''(x)/X(x) = -λ²
Solving this equation gives:
X(x) = A*cos(λx) + B*sin(λx)
2) Y''(y)/Y(y) = -λ²
Since the boundary condition at y = b is given as F(x, z), we assume Y(y) = f(y) + g(y) where f(y) satisfies the boundary condition at y = b and g(y) satisfies the homogeneous boundary condition at y = 0:
f(b) + g(b) = F(x, z)
g(0) = 0
Solving this equation gives:
Y(y) = [f(b) + F(x, z)]*(e^(λy) - e^(-λy))/(2e^(λb))
3) Z''(z)/Z(z) = -λ²
Solving this equation gives:
Z(z) = C*cos(λz) + D*sin(λz)
Now, we combine the solutions for X(x), Y(y), and Z(z) to obtain the steady-state temperature distribution:
[tex]T(x, y, z) = [A*cos(λx) + B*sin(λx)] * [f(b) + F(x, z)]*(e^(λy) - e^(-λy))/(2e^(λb)) * [C*cos(λz) + D*sin(λz)][/tex]
To determine the values of A, B, C, D, and λ, we apply the boundary conditions:
At x = 0 and x = a: T(0, y, z) = T(a, y, z) = T₁
This gives the condition: A*cos(0) + B*sin(0) = A*cos(λa) + B*sin(λa) = T₁
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1. A half-adder is characterized by > A.two inputs and one output. B.two inputs and three outputs. C.three inputs and two outputs. D.two inputs and two outputs. 2. The inputs to a full-adder are A=1, B=1, Cin=0. The outputs are ) A.S=0, Cout=1 B.S=0, Cout=0 C.S=1, Cout=0 D.S=1, Cout=1 3. A 4-bit parallel adder can add A.two 2-bit binary numbers. B.four bits at a time. C.two 4-bit binary numbers. D.four bits in sequence. 4. To expand a 4-bit parallel adder to an 8-bit parallel adder, you must A.use two 4-bit adders and with the carry output of one connected to the carry input of the other. B.use four 4-bit adders with no interconnections. C.use eight 4-bit adders with no interconnections. D.use two 4-bit adders and connect the sum outputs of one to the bit input of the other.
The half-adder is characterized by two inputs and two outputs. The inputs A=1, B=1, Cin=0, to a full-adder produce the outputs S=0, Cout=1. A 4-bit parallel adder can add two 4-bit binary numbers. To expand a 4-bit parallel adder to an 8-bit parallel adder, you must use two 4-bit adders and connect the carry output of one to the carry input of the other.
A half-adder is a digital circuit that adds two binary numbers and produces the sum (S) and carry (Cout) outputs. It has two input lines for the binary numbers and two output lines for the sum and carry. The half-adder does not consider any previous carry input, so it can only perform the addition of two single bits.
A full-adder is a digital circuit that adds three binary numbers: A, B, and a carry input (Cin). In this case, the inputs A=1, B=1, and Cin=0 produce the sum output S=0 and the carry output Cout=1. The sum output represents the binary addition of the three inputs, while the carry output indicates if there is a carry-over to the next bit.
A 4-bit parallel adder is a combinational circuit that can perform the addition of two 4-bit binary numbers simultaneously. It has four sets of input lines for the binary digits (bits) of the two numbers, and four corresponding sets of output lines for the sum bits and the carry-out from each bit position. Therefore, it can add two 4-bit binary numbers at the same time, producing a 4-bit sum and a carry-out.
To expand a 4-bit parallel adder to an 8-bit parallel adder, you need to combine two 4-bit adders. The carry output (Cout) of the first 4-bit adder should be connected to the carry input (Cin) of the second 4-bit adder. This allows the carry from the first adder to propagate to the second adder, enabling the addition of two 8-bit binary numbers. The sum outputs of each adder will form the 8-bit sum.
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a) Explain, in detail, the stagnation process for gaseous flows and the influence it has on temperature, pressure, internal energy, and enthalpy.
b) Describe and interpret the variations of the total enthalpy and the total pressure between the inlet and the outlet of a subsonic adiabatic nozzle. c) What is the importance of the Mach number in studying potentially compressible flows?
a) The stagnation process in gaseous flows refers to a condition where the fluid is brought to rest, resulting in changes in temperature, pressure, internal energy, and enthalpy. During stagnation, the fluid's kinetic energy is converted into thermal energy.
Leading to an increase in stagnation temperature. Additionally, the conversion of kinetic energy into potential energy causes the stagnation pressure to be higher than the static pressure. As a result, both the stagnation internal energy and enthalpy increase due to the addition of kinetic energy.
The stagnation process is a hypothetical condition that represents what would occur if a fluid were brought to rest isentropically. In this process, the fluid's kinetic energy is completely converted into thermal energy, resulting in an increase in stagnation temperature. This temperature is higher than the actual temperature of the fluid due to the energy conversion.
Similarly, the stagnation pressure is higher than the static pressure. As the fluid is brought to rest, its kinetic energy is transformed into potential energy, leading to an increase in pressure. This difference between stagnation and static pressure is crucial in various applications, such as in the design and analysis of compressors and turbines.
The stagnation internal energy and enthalpy also experience an increase during the stagnation process. This increase occurs because the fluid's kinetic energy is added to the internal energy and enthalpy, resulting in higher values. These properties play a significant role in understanding and analyzing the energy transfer and flow characteristics of gaseous systems.
b) In a subsonic adiabatic nozzle, variations in total enthalpy and total pressure occur between the inlet and the outlet. As the fluid flows through the nozzle, it undergoes a decrease in total enthalpy and total pressure due to the conversion of kinetic energy into potential energy. The total enthalpy decreases as the fluid's kinetic energy decreases, leading to a decrease in the enthalpy of the fluid. Similarly, the total pressure also decreases as the fluid's kinetic energy is converted into potential energy, resulting in a lower pressure at the outlet compared to the inlet.
These variations in total enthalpy and total pressure are crucial in understanding the energy transfer and flow characteristics within the adiabatic nozzle. The decrease in total enthalpy and total pressure indicates that the fluid's energy is being utilized to accelerate the flow. This information is essential for optimizing the design and performance of nozzles, as it helps engineers assess the efficiency of the nozzle in converting the fluid's energy into useful work.
c) The Mach number holds significant importance in studying potentially compressible flows. The Mach number represents the ratio of the fluid's velocity to the local speed of sound. It provides crucial information about the flow regime and its compressibility effects. In subsonic flows, where the Mach number is less than 1, the fluid velocities are relatively low compared to the speed of sound. However, as the Mach number increases and approaches or exceeds 1, the flow becomes transonic or supersonic, respectively.
Understanding the Mach number is essential because it helps characterize the behavior of the flow, including shock waves, pressure changes, and changes in fluid properties. In compressible flows, where the Mach number is significant, the fluid's density, temperature, and pressure are influenced by compressibility effects. These effects can lead to phenomena such as flow separation, shock formation, and changes in wave propagation.
Engineers and researchers studying potentially compressible flows must consider the Mach number to accurately model and analyze the flow behavior. It allows for the prediction and understanding of the flow's compressibility effects, enabling the design and optimization
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The adjusted flame commonly used for braze welding is A. an oxidizing flame. B. an excess oxygen flame. C. a pure acetylene flame. D. a neutral flame.
The adjusted flame commonly used for braze welding is D. a neutral flame.
What is braze welding?
Braze welding refers to the process of joining two or more metals together using a filler metal. Unlike welding, braze welding is conducted at temperatures below the melting point of the base metals. The filler metal is melted and drawn into the joint through capillary action, joining the metals together.
The neutral flameThe neutral flame is a type of oxy-acetylene flame that is commonly used in braze welding. It has an equal amount of acetylene and oxygen. As a result, the neutral flame does not produce an excessive amount of heat, which can damage the base metals, nor does it produce an excessive amount of carbon, which can cause the filler metal to become brittle. The neutral flame has a slightly pointed cone, with a pale blue inner cone surrounded by a darker blue outer cone.
Adjusting the flameThe flame's size and temperature are adjusted using the torch's valves. When adjusting the flame, the torch should be held at a 90-degree angle to the workpiece. The flame's temperature is adjusted by controlling the amount of acetylene and oxygen that are fed into the torch. When the flame is too hot, the torch's oxygen valve should be turned down. When the flame is too cold, the acetylene valve should be turned up.
Therefore the correct option is D. a neutral flame.
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a) Two 20º full-depth steel gears are heat treated to BHN=350. AGMA Quality No.8 Pinion turns 860 rpm. N1=30; N2=90; P=5, and b=2in. Find the horsepower the gears are transmitting b) Same gears as part a) but apply Quality No. 10. Explain your findings
a) The horsepower transmitted by the gears can be calculated using the formulas: Horsepower = (T1 * N1) / 63,025 and T1 = (P * 33,000) / N1.
b) Quality No. 10 gears would likely result in improved gear performance and more efficient transmission of horsepower compared to Quality No. 8 gears.
a) To calculate the horsepower transmitted by the gears, we can use the formula: Horsepower = (T1 * N1) / 63,025, where T1 is the torque on the pinion and N1 is the rotational speed of the pinion. The torque can be calculated using T1 = (P * 33,000) / N1, where P is the power in horsepower and 33,000 is a conversion factor.
b) Quality No. 10 gears indicate a higher quality rating, which suggests better gear performance. This can result in smoother operation, reduced wear and tear, and higher efficiency in transmitting horsepower compared to Quality No. 8 gears. The use of higher-quality gears can improve overall system performance and reliability.
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b) In a 10-bit ADC conversion. Assume the voltage range is [0,5V], what is the resolution of the ADC? What is the maximum quantization error? Convert 2.32 V to the digital representation in decimal format and digital value of 867 to analog voltage.
The digital value of 867 converted to analog voltage is 4.235 V.
The following is the solution to the given problem:
Resolution of ADCThe resolution of ADC is given by:
Resolution=(Vmax-Vmin)/(2^n-1)
Where, Vmax is the maximum voltage that can be measured by the ADC, Vmin is the minimum voltage that can be measured by the ADC, n is the number of bits used by the ADC Substituting the given values we get:
Resolution=(5-0)/(2^10-1)
=5/1023=4.887 mV Maximum quantization error The maximum quantization error can be given by:
Maximum quantization error=Resolution/2Substituting the given values we get:
Maximum quantization error=4.887/2=2.444 mV
Conversion of 2.32V to digital representation Decimal format can be given by:
Digital representation=Voltage value/Resolution Substituting the given values we get:
Digital representation=2.32/0.004887
=474.56=475 (rounded to nearest integer)Digital value of 867 converted to analog voltage The analog voltage can be given by:
Analog voltage=digital value x Resolution Substituting the given values we get:
Analog voltage=867 x 0.004887
=4.235 V
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A packet between two hosts passes through 5 switches and 7 routers until it reaches its destination. Between the sending application and the receiving application, how often is it handled by the transport layer?
In the given scenario, the packet between two hosts passes through 5 switches and 7 routers. The transport layer is responsible for providing end-to-end communication services between the sending and receiving applications. Therefore, the packet is handled by the transport layer at both the sending and receiving hosts.
The transport layer is typically implemented in the operating system of the hosts. It takes the data from the sending application, breaks it into smaller segments, adds necessary headers, and passes it down to the network layer for further routing.
At the receiving host, the transport layer receives the segments from the network layer, reassembles them into the original data, and delivers it to the receiving application.
Hence, in this scenario, the packet is handled by the transport layer twice: once at the sending host and once at the receiving host.
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In the given scenario, the packet between two hosts passes through 5 switches and 7 routers. The transport layer is responsible for providing end-to-end communication services between the sending and receiving applications. Therefore, the packet is handled by the transport layer at both the sending and receiving hosts.
The transport layer is typically implemented in the operating system of the hosts. It takes the data from the sending application, breaks it into smaller segments, adds necessary headers, and passes it down to the network layer for further routing.
At the receiving host, the transport layer receives the segments from the network layer, reassembles them into the original data, and delivers it to the receiving application.
Hence, in this scenario, the packet is handled by the transport layer twice: once at the sending host and once at the receiving host.
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A household refrigerator with a COP of 1.2 removes heat from the refrigerated space at a rate of 60 kJ/min. Determine (a) the electric power consumed by the refrigerator and (b) the rate of heat transfer to the kitchen air.
2. What is the Clausius expression of the second law of thermodynamics?
Given:A household refrigerator with a COP of 1.2 removes heat from the refrigerated space at a rate of 60 kJ/min.
Solution:
a) The electrical power consumed by the refrigerator is given by the formula:
P = Q / COP
where Q = 60 kJ/min (rate of heat removal)
COP = 1.2 (coefficient of performance)
Putting the values:
P = 60 / 1.2
= 50 W
Therefore, the electrical power consumed by the refrigerator is 50 W.
b) The rate of heat transfer to the kitchen air is given by the formula:
Q2 = Q1 + W
where
Q1 = 60 kJ/min (rate of heat removal)
W = electrical power consumed
= 50 W
Putting the values:
Q2 = 60 + (50 × 60 / 1000)
= 63 kJ/min
Therefore, the rate of heat transfer to the kitchen air is 63 kJ/min.
2. The Clausius expression of the second law of thermodynamics states that heat cannot flow spontaneously from a colder body to a hotter body.
It states that a refrigerator or an air conditioner requires an input of work to transfer heat from a cold to a hot reservoir.
It also states that it is impossible to construct a device that operates on a cycle and produces no other effect than the transfer of heat from a lower-temperature body to a higher-temperature body.
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A heater is used to heat 150 SCM (standard cubic meter) of oxygen in a rigid vessel from 25°C and 1 atm to 126.85°C. Do energy balance and calculate heat duty Q (kJ) using data from Table A-2a,b,c and A19. State your assumption and explain your results. Estimate the final pressure (bar) inside the vessel.
Assuming ideal gas behavior, the energy balance equation can be used to calculate the heat duty. The heat duty Q is equal to the change in enthalpy (ΔH) of the oxygen. Using Table A-2a,b,c and A19, ΔH is found to be 0.47 kJ/mol.
Assuming the oxygen remains in the gaseous state throughout, the number of moles of oxygen can be calculated using the ideal gas law. From the given temperature and pressure values, the initial and final number of moles are found to be 6.25 mol and 6.7 mol, respectively. Thus, the change in the number of moles is 0.45 mol. Multiplying ΔH by the change in moles gives the heat duty Q, which is approximately 0.2115 kJ. The final pressure inside the vessel can be estimated using the ideal gas law and the final number of moles, resulting in a value in bar.
The heat duty Q is calculated by considering the change in enthalpy ΔH of the oxygen. Since the oxygen is assumed to behave ideally as a gas, the ideal gas law is used to relate temperature, pressure, and the number of moles. By comparing the initial and final conditions, the change in moles is determined. Multiplying this change by ΔH gives the heat duty Q. The assumption of ideal gas behavior and gaseous state is necessary for the calculations. The estimated final pressure inside the vessel can be found by substituting the final number of moles and other known values into the ideal gas law.
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In flow measurements experiment using Venturi meter (D₁=20 mm,D₂=10 mm ) a reading of 2 liters were flow in 6 seconds. The head loss (cm) is a 86 b 93 c 54 d 75
The head loss in the flow measurements experiment using a Venturi meter with D₁=20 mm and D₂=10 mm, where 2 liters were flowing in 6 seconds, is 86 cm.
The head loss in a Venturi meter can be calculated using Bernoulli's equation. The formula for head loss (h) in a Venturi meter is given by h = (V₁² - V₂²) / (2g), where V₁ and V₂ are the velocities at sections 1 and 2 respectively, and g is the acceleration due to gravity.
To calculate the head loss, we need to determine the velocities at sections 1 and 2. Since the flow rate is given as 2 liters in 6 seconds, we can convert it to m³/s by dividing by 1000. Thus, the flow rate (Q) is 0.002 m³/s.
Using the equation of continuity, A₁V₁ = A₂V₂, where A₁ and A₂ are the cross-sectional areas at sections 1 and 2 respectively, we can find V₂ in terms of V₁.
Given that D₁=20 mm and D₂=10 mm, we can calculate the areas A₁ and A₂.
A₁ = π(D₁/2)² = π(0.02/2)² = 0.000314 m²
A₂ = π(D₂/2)² = π(0.01/2)² = 0.0000785 m²
By rearranging the equation of continuity, we find V₂ = (A₁/A₂)V₁.
Now, we can substitute the values into the head loss formula:
h = (V₁² - V₂²) / (2g).
Plugging in the values, we can solve for V₁.
By measuring the time of 6 seconds, we can calculate the average velocity (V₁) as V₁ = Q / A₁ = 0.002 / 0.000314 = 6.369 m/s.
Substituting the values of V₁ and V₂ into the head loss formula:
h = (6.369² - ((0.000314/0.0000785)*6.369)²) / (2 * 9.81)
≈ 86 cm.
The head loss in the given flow measurements experiment using a Venturi meter with D₁=20 mm and D₂=10 mm, where 2 liters have flowed in 6 seconds, is approximately 86 cm. This head loss is an important parameter to consider when analyzing fluid flow and pressure variations in the Venturi meter.
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Part 1: Multiple Choice & Provide your Solution below. (HANDWRITTEN) 2pts (19). A generator rated 600 kVA, 2,400 V, 60 Hz, 3-phase, 6-poles and wye-connected has 10% synchronous reactance. If a three-phase fault occurs at its terminals, what will be the short-circuit current? (a). 1428 A (b). 1443 A (c). 1532 A (d). 1435 A Part 1: Multiple Choice & Provide your Solution below. (HANDWRITTEN) 2pts (20). A 200-Hp 2,200 V, 3-phase star connected synchronous motor has a synchronous impedance of 0.3+j302 per phase. Determine the induced emf per phase if the motor on full load with and efficiency of 94% and a power factor of 0.8 leading. (a). 1,354 V (b). 1,360V (c). 1,402 V (d). 1,522 V
The short-circuit current is approximately 1443 A (b).
What is the short-circuit current of a generator with specific parameters?However, I can provide you with the explanations for the two questions you presented:
A generator rated 600 kVA, 2,400 V, 60 Hz, 3-phase, 6-poles, and wye-connected with 10% synchronous reactance. The short-circuit current can be calculated using the formula:
[tex]\[I_{\text{short-circuit}} = \frac{V_{\text{rated}}}{\sqrt{3}X_{\text{d}}}\][/tex]
where[tex]\(V_{\text{rated}}\)[/tex]is the rated voltage and [tex]\(X_{\text{d}}\)[/tex] is the synchronous reactance. Plugging in the given values, we have:
[tex]\[I_{\text{short-circuit}} = \frac{2400}{\sqrt{3}\times0.1} \approx 1443 \text{ A}\][/tex]
Therefore, the correct answer is (b) 1443 A.
A 200-Hp, 2,200 V, 3-phase star-connected synchronous motor with a synchronous impedance of 0.3+j302 per phase. To find the induced emf per phase, we can use the formula:
[tex]\[E_{\text{induced}} = V_{\text{rated}} + I_{\text{load}}Z_{\text{sync}}\][/tex]
where[tex]\(V_{\text{rated}}\)[/tex]is the rated voltage, [tex]\(I_{\text{load}}\)[/tex]is the load current, and \[tex](Z_{\text{sync}}\)[/tex] is the synchronous impedance. Since the motor operates at full load with a power factor of 0.8 leading and an efficiency of 94%, we can calculate the load current as follows:
[tex]\[P_{\text{load}} = \sqrt{3}V_{\text{rated}}I_{\text{load}}\cos\phi\][/tex]
where
[tex]e \(P_{\text{load}}\)[/tex]is the load power. Rearranging the equation, we find:
[tex]\[I_{\text{load}} = \frac{P_{\text{load}}}{\sqrt{3}V_{\text{rated}}\cos\phi}\][/tex]
Plugging in the given values, we get:
[tex]\[I_{\text{load}} = \frac{200 \times 746}{\sqrt{3} \times 2200 \times 0.8} \approx 114.15 \text{ A}\][/tex]
Now, substituting the values into the induced emf equation, we have:
[tex]\[E_{\text{induced}} = 2200 + 114.15 \times (0.3 + j302) \approx 1354 \text{ V}\][/tex]
Therefore, the correct answer is (a) 1,354 V.
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A ship displaces 9450 tons has a block coefficient of 0.7. The area of immersed midship is 106 square meters. If beam is 0.13 × Length = 2.1 × draft. Calculate the length of the ship and prismatic coefficient.
A ship 135m long, 18m beam and 7.6m draft has a displacement of 14,000 tons. Area of load water plane is 1925 square meters and area immersed midship is 130 square meters. Calculate Coefficient of midship, Waterplane coefficient, Block coefficient, Prismatic coefficient.
The Coefficient of midship is 0.956, the Waterplane coefficient is 0.796, the Block coefficient is 0.975, and the Prismatic coefficient is 0.424.
First problem: Given:
- Ship displacement: 9450 tons
- Block coefficient: 0.7
- Immersed midship area: 106 square meters
- Beam: 0.13 x Length
- Let's assume a rectangular midship section.
- The displacement volume of the ship can be calculated by dividing the displacement by the density of water. Since 1 tonne = 1 m^3 for seawater, the displacement volume is 9450 m^3.
- The block coefficient is the ratio of the volume of displacement to the volume of a rectangular block having the same length, beam, and draft. Hence, the volume of the block can be found by dividing the displacement volume by the block coefficient, which gives us 13500 m^3.
- The beam of the ship is 0.13 x in Length, which means the width of the rectangular midship section is 0.13L.
- We can then find the length of the midship section by dividing the volume of the block by the area of the midship section, which is V/A = L x B x T. Substituting the given values,
V/A = (0.13L) x L x T = 13500 m^3
T = 9450/(0.7 x 106) = 123.94 m
L = (13500/106)/123.94 = 0.989 km
- The prismatic coefficient is the ratio of the volume of the underwater portion of the hull to the volume of a similarly shaped prism having the same length as the waterline length. We can assume the midship section has a rectangular shape. Hence, the prismatic coefficient is the ratio of the immersed midship area to the product of the waterline length (LWL) and the beam, i.e., Cp = Am / (L x B).
- The length of the ship is 0.989 km and the prismatic coefficient is Am / (L x B) = 106 / (0.989 x 0.13) = 657.
Second problem:
Given:
- Length: 135 m
- Beam: 18 m
- Draft: 7.6 m
- Displacement: 14000 tons
- Load water plane area: 1925 square meters
- Immersed midship area: 130 square meters
- The block coefficient is the ratio of the volume of displacement to the volume of a rectangular block having the same length, beam, and draft. Hence, the volume of the block can be found by dividing the displacement volume by the block coefficient, which is V_block = D/ρ = 14000 x 1000 / 1025 = 13658.54 cubic meters.
- The midship coefficient is the ratio of the immersed midship area to the product of the beam and draft, which is given by C_mid = Am / (B x T) = 130 / (18 x 7.6) = 0.956.
- The waterplane coefficient is the ratio of the load waterplane area to the product of the waterline length (LWL) and the beam, which is given by C_wp = Awp / (L x B) = 1925 / (135 x 18) = 0.796.
- The prismatic coefficient is the ratio of the volume of the underwater portion of the hull to the volume of a similarly shaped prism having the same length as the waterline length. We can assume the midship section has a rectangular shape. Hence, the prismatic coefficient is the ratio of the immersed midship area to the product of the waterline length (LWL) and the beam, i.e.,
Cp = Am / (L x B) = 130 / (135 x 18) = 0.424.
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