Engineering College

## Answers

**Answer 1**

a) **The mixture **is lean since the given equivalence ratio (q) is 0.8, which is less than the stoichiometric value.

b) The balanced chemical equation describing the combustion process is: C8H18 + 12.5O2 + (q × (79/21) × 3.76)N2 → 8CO2 + 9H2O + (q × 3.76)N2, where q = 0.8.

c) The mole fractions of each gas in the exhaust are: CO2 = 32/1505, H2O = 36/493, and N2 = 1127/1505.

d) The actual air flow rate in kg/sec is 174.42 kg/sec.

The given equivalence ratio is q = 0.8. Since the** air-fuel ratio** is less than the **stoichiometric **value, so the mixture is lean. The balanced chemical equation for the combustion process is:

C8H18 + 12.5O2 + (q × (79/21) × 3.76)N2 → 8CO2 + 9H2O + (q × 3.76)N2

Here, q = 0.8.

Air consists of 79% nitrogen and 21% oxygen. Therefore, the mole fractions of O2 and N2 in the mixture are (0.21q × 3.76) and (0.79q × 3.76), respectively. The mass fraction of CO2 in the exhaust is 8/29, and the mass fraction of H2O in the exhaust is 9/29. Calculating the mole fractions:

nCO2 = (8/(29 × 44)) × (1/(1 + 9/8)) = 32/1505nH2O = (9/(29 × 18)) × (1/(1 + 9/8)) = 36/493

**Mole fraction **of N2 in exhaust gas = 1 - mole fraction of CO2 - mole fraction of H2O

nN2 = 1 - (32/1505) - (36/493) = 1127/1505

The fuel flow rate is given as ṁfuel = 1.5 × 10^9 g/sec. The stoichiometric mass of air required for the combustion of 1 g of fuel is calculated as:

ma,stoch = (3.76 × 28.84)/12.5 = 8.6 g

The mass of air supplied per second is:

= ṁair = ṁfuel / ma,stoch = (1.5 × 10^9) / 8.6 = 174,418,604.7 g/sec = 174.42 kg/sec

Therefore, the main answer is the given mixture supplied to the 4-cylinder laboratory test engine is lean (a), and the balanced chemical equation for the combustion process with **isooctane fuel **and an air equivalence ratio of 0.8 is C8H18 + 12.5O2 + (0.8 × (79/21) × 3.76)N2 → 8CO2 + 9H2O + (0.8 × 3.76)N2.

The mole fractions of the gases in the exhaust are CO2 = 32/1505, H2O = 36/493, and N2 = 1127/1505 (c), while the **actual air flow **rate is 174.42 kg/sec (d).

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## Related Questions

The following design data apply to an inward flow radial turbine:

Overall efficiency 75%

Net head across the turbine 6 m

Power output 128 kW

The runner tangential velocity 10.6 m/s

Flow velocity 4 m/s

Runner rotational speed 235 rpm

Hydraulic losses 18% of energy available

Calculate the inlet guide vane angle, the inlet angle of the runner vane, the runner diameter at inlet, and height of the runner at inlet. Assume that the discharge is radial.

### Answers

To calculate the required design **parameters** for the inward flow radial turbine, we'll use the given data and apply relevant formulas. These values are calculated based on the given design data, such as overall efficiency, net head across the turbine, power output, runner tangential velocity, flow velocity, runner rotational speed, and hydraulic losses.

Inlet **Guide** Vane (IGV) Angle:

The IGV angle (α) determines the direction of the flow at the runner **inlet**. It can be calculated using the formula:

α = tan^(-1)((U - V) / W), where U is the runner **tangential** velocity (10.6 m/s), V is the flow velocity (4 m/s), and W is the net head across the **turbine** (6 m).

**Plugging** in the values, we get:

α = tan^(-1)((10.6 - 4) / 6)

= 39.23 degrees.

Inlet Angle of **Runner** Vane:

The inlet angle (β1) of the runner **vane** is related to the IGV angle (α) by the formula:

β1 = α + tan^(-1)((1 - ηh) / ηh), where ηh is the **hydraulic** losses (18% or 0.18).

**Substituting** the values, we get:

β1 = 39.23 + tan^(-1)((1 - 0.18) / 0.18)

= 48.68 degrees.

Runner **Diameter** at Inlet:

The **runner** diameter at the inlet can be calculated using the formula:

D1 = (2P / (πNρUcos(β1))), where P is the **power** output (128 kW), N is the runner **rotational** speed (235 rpm), ρ is the **density** of the fluid, and cos(β1) is the cosine of the inlet angle of the runner vane.

As the density is not given, we cannot provide a specific numerical value for the diameter.

Height of Runner at Inlet:

The height (H1) of the runner at the inlet is related to the runner diameter (D1) and the flow **velocity** (V) by the formula:

H1 = (D1V) / U

As the diameter is not provided, we cannot provide a specific numerical value for the **height**.

In conclusion, the **calculated** values are as follows:

Inlet Guide Vane (IGV) Angle: 39.23 **degrees**

Inlet Angle of Runner Vane: 48.68 degrees

Runner Diameter at Inlet: N/A (density **information** missing)

Height of Runner at Inlet: N/A (diameter information missing)

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(a) A 50mm thick carpet made of polypropylene is burning in front of the doorway to a room that is on fire. If the carpet is receiving 12kW/m2 in the form of radiant heat from the room, would the carpet continue to burn if the door was closed (no radiation)?

20% of the heat energy released by burning is transferred back to the carpet by radiation;

The carpet is losing heat only due to radiation and steady state conduction between the carpet surface and the floor beneath it.

The carpet is a black-body radiator with a black-body temperature of 650C

The temperature at the floor beneath the carpet is 30C

Thermal conductivity co-efficient of the carpet material is 0.18 W/m.K

. Lv for carpet is 1.8 MJ/kg

Critical mass flux is 0.015 kg m-2s-1

### Answers

Based on the calculations, the carpet would not continue to burn if the door was closed and there was no radiation. The heat **energy **lost by the carpet due to radiation and conduction is greater than the heat energy received from the room..

To determine if the carpet would continue to burn, we need to compare the heat energy received by the carpet from the room (12 kW/m²) with the heat energy lost by the carpet due to radiation and conduction.

First, let's calculate the heat energy lost by the carpet due to **radiation**. We know that 20% of the heat energy released by burning is transferred back to the carpet by radiation. Therefore, the heat energy lost by radiation is:

Heat energy lost by radiation = 0.20 * 12 kW/m² = 2.4 kW/m²

Next, let's calculate the heat energy lost by the carpet due to conduction. We can use the steady-state **conduction **equation:

Heat energy lost by conduction = (Thermal conductivity) * (Area) * (Temperature difference / Thickness)

The temperature difference is the difference between the carpet temperature and the floor temperature:

Temperature difference = (650°C - 30°C) = 620°C

Now, we can calculate the heat energy lost by conduction:

Heat energy lost by conduction = (0.18 W/m·K) * (1 m²) * (620°C / 0.05 m) = 2232 kW/m²

Therefore, the total heat energy lost by the carpet due to radiation and conduction is:

Total **heat **energy lost = Heat energy lost by radiation + Heat energy lost by conduction

= 2.4 kW/m² + 2232 kW/m²

= 2234.4 kW/m²

Now, let's compare the heat energy lost with the heat energy received. If the heat energy lost is greater than or equal to the heat energy received, the carpet will not continue to burn.

Heat energy received = 12 kW/m²

Since 2234.4 kW/m² is greater than 12 kW/m², the carpet would not continue to burn if the door was closed and there was no radiation.

Based on the calculations, the carpet would not continue to burn if the door was closed and there was no radiation. The heat energy lost by the carpet due to radiation and conduction is greater than the heat energy received from the room.

Therefore, the **carpet **would eventually cool down and stop burning.

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When designing a composite structure these a preliminary design methodology you need to follow, list two specification involved in a design process and two specific application requirements involved

### Answers

When designing a composite **structure**, there is a preliminary design methodology that needs to be followed. Two specifications involved in the design process are design loads and material** properties.**

Design Loads are the forces that an object experiences throughout its lifespan. These loads include gravitational, thermal, pressure, wind, and more. Knowing the design loads helps engineers determine how much force a structure can withstand before it **fails.**

Material Properties refers to the characteristics of materials that are used in composite structures. Material properties are divided into two categories: mechanical and **physical.**

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for double output if Vo1 and Vo2 are different.

what is the general Vo=? formula

Differential amplifier if vio #V02, Does Vo = Advd just ? or Vo= Voide + Advd + Acvc

### Answers

In a differential **amplifier** configuration, where the inputs are denoted as Vio and V02, the general formula for the output voltage (Vo) can be expressed as:

Vo = Ad * (Vio - V02) + Vo_CM

Where:

Ad represents the differential gain of the amplifier, which amplifies the voltage difference between the two inputs.

Vio is the voltage applied to the **non-inverting** input.

V02 is the voltage applied to the inverting input.

Vo_CM is the common-mode output voltage, which represents any output offset that is not directly related to the input differential voltage.

The common-mode output voltage, Vo_CM, can include contributions from various factors such as power supply noise, component mismatches, and amplifier imperfections.

Therefore, the general formula for the output **voltage **of a differential amplifier includes both the differential gain term (Ad * (Vio - V02)) and the common-mode term (Vo_CM).

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heat transfer

Is the fin efficiency the same throughout the length of the fin? What is the efficiency of fins used in practice? At what efficiency the fins cannot be justified economically?

### Answers

The fin efficiency is not uniform along the length of the fin, and the efficiency of fins used in practice varies. Fins become** economically** unjustifiable when their efficiency is low and the cost of installation and maintenance outweighs the heat transfer benefits.

What factors influence the efficiency of fins used for heat transfer, and when do fins become economically unjustifiable?

The efficiency of a fin refers to its ability to transfer heat effectively from the surface it is attached to. It is typically defined as the ratio of actual heat transfer through the fin to the **maximum** possible heat transfer if the entire fin was at the base temperature.

The fin efficiency is not uniform along the length of the fin because heat transfer mechanisms, such as conduction, convection, and radiation, vary with position. The efficiency is usually highest near the base of the fin where the temperature difference between the fin and the surrounding medium is larger. As we move towards the tip of the fin, the temperature difference **decreases**, resulting in reduced efficiency.

In practical applications, the efficiency of fins depends on various factors such as fin geometry, material properties, airflow conditions, and surface temperature. Typical fin efficiencies range from 70% to 90%, with variations depending on the specific design and operating parameters.

Fins become economically unjustifiable when their efficiency is low and the cost of installing and maintaining them outweighs the benefits gained in terms of **enhanced** heat transfer. The specific efficiency threshold at which fins are deemed uneconomical can vary depending on the application, cost considerations, and desired heat transfer improvements.

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Consider the following equations in which X₁, X2,..., Xn, are variables, and a₁, a2,..., an, are general coefficients or mathematical operators: Draw a block diagram for given equation, identifying all blocks, inputs, and outputs. Xn = a1x1 + a₂x2 +an-1Xn-1

### Answers

The **block diagram** of the given equation helps in understanding the structure of the equation and the **operations** involved.

The given **equation** is as follows:

Xn = a1x1 + a2x2 + ... + an-1Xn-1

This equation can be represented using a block diagram as shown below:

Block diagram of the given equation

The block diagram of the given equation consists of the following blocks:

**Multiplier block**: The input variables x1, x2, xn-1, etc. are multiplied by the respective coefficients a1, a2, an-1, etc. using the multiplier blocks. The output of each multiplier block is the product of the input and the coefficient.

**Adder block**: The outputs of the multiplier blocks are then added using the adder block. The output of the adder block is the value of Xn.

Inputs: The input variables x1, x2, xn-1 are the inputs to the multiplier blocks. The coefficients a1, a2, an-1 are also inputs to the multiplier blocks.

Outputs: The output of the block diagram is the value of Xn which is obtained at the output of the adder block.

The number of multiplier blocks in the block diagram is equal to the number of variables (x1, x2, xn-1) in the equation. The adder block is common to all multiplier blocks, and it adds the products obtained from all the multiplier blocks.

The block diagram of the given equation helps in understanding the structure of the equation and the operations involved.

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estimate the bare cost of providing and installing 12 sliding windows 5ft x 4ft including screen. assume vinyl clad, premium with double insulating glass. the job will be in miami florida. cci for Miami Florida: materials 97.9%; labor 69.5%; total 90.7%.

### Answers

The bare cost of providing and installing 12 sliding windows 5ft x 4ft including screen assuming vinyl clad, premium with double insulating glass in Miami, Florida is approximately **$9,120.38**.

The calculation of the bare cost of providing and installing 12 sliding windows 5ft x 4ft including screen, assuming vinyl clad, premium with double insulating glass in Miami, Florida can be done in the following way. The material cost is found by calculating the area of one window which is 5ft x 4ft = 20 sq.ft. For **12 windows**, the area would be 240 sq.ft. The vinyl material cost is assumed to be $15 per sq.ft. So, the material cost would be 240 x 15 = $3,600.

The** labor cost **is calculated by taking 69.5% of the material cost. Then, the total cost is found by adding the material and labor costs. The total cost is equal to 90.7% of the sum of the material and labor costs.

Let X be the cost of materials. Therefore; Labor cost = 69.5/100 × X Total cost = 90.7/100 × (X + 69.5/100 × X) Total cost = 90.7/100 × (1 + 69.5/100) × X Total cost = 90.7/100 × 1.695 × X **Total cost = 153.8125/100 × X**

Using this **formula**, the bare cost of providing and installing 12 sliding windows 5ft x 4ft including screen assuming vinyl clad, premium with double insulating glass in Miami, Florida is approximately $9,120.38.

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Using Simulink toolbox in MATLAB, develop and analyse a design of PID controller for Knee joint angle system for prosthetic legs. For your design, please provide those items as follows:

Block diagram of each part of the systems and controllers

### Answers

The steps to **design **and examine a **PID controller **for a knee joint angle system in Simulink are:

**Plant Modeling** and sensor ModelingReference Input and error CalculationPID Controller Design and actuator ModelingPlant and Controller Integration and simulation and Analysis

What is the MATLAB?

In the above, the **reference **input tells the knee joint how far to bend. - The error block figures out the difference between what the knee should be doing and what it's actually doing.

Therefore, The actuator is like a **messenger **that delivers the message to the knee joint to make it do what it should. The plant is like the knee joint, and the sensor checks how bent the knee is.

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An aircraft is flying at a speed of 480 m/s. This aircraft used the simple aircraft air conditioning cycle and has 10 TR capacity plant as shown in figure 4 below. The cabin pressure is 1.01 bar and the cabin air temperature is maintained at 27 °C. The atmospheric temperature and pressure are 5 °C and 0.9 bar respectively. The pressure ratio of the compressor is 4.5. The temperature of air is reduced by 200 °C in the heat exchanger. The pressure drop in the heat exchanger is neglected. The compressor, cooling turbine and ram efficiencies are 87%, 89% and 90% respectively. Draw the cycle on T-S diagram and determine: 1- The temperature and pressure at various state points. 2- Mass flow rate. 3- Compressor work. 4- COP.

### Answers

1- The** temperature** and **pressure** at various state points:

State 1: Atmospheric conditions - T1 = 5°C, P1

= 0.9 bar

State 2: Compressor exit - P2 = 4.5 * P1, T2 is determined by the compressor efficiency

State 3: Cooling **turbine exit** - P3 = P1, T3 is determined by the temperature reduction in the heat exchanger

State 4: Ram air inlet - T4 = T1,

P4 = P1

State 5: Cabin conditions - T5 = 27°C,

P5 = 1.01 bar

2-** Mass flow rate**:

The mass flow rate can be calculated using the equation:

Mass flow rate = Cooling capacity / (Cp × (T2 - T3))

3- **Compressor work**:

Compressor work can be calculated using the equation:

Compressor work = (h2 - h1) / Compressor efficiency

4- Coefficient of Performance (COP):

COP = Cooling capacity / Compressor work

Please note that specific values for cooling capacity and Cp (specific heat at constant pressure) are required to calculate the above parameters accurately.

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An aircraft with a turbojet engine flies at Mach number M = 0.85 at an altitude where the ambient air temperature is 216.7 K and pressure is 18.75 kPa. A percentage of the compressor air flow is bled and used to cool the turbine blades – this cooling air did not contribute to the engine propulsion. Temperature sensors in the engine show the compressor outlet, turbine inlet and turbine outlet temperatures respectively to be 687 K, 1700 K and 1261 K, whilst turbine outlet pressure is 227 kPa. If the engine's specific thrust (T/ma) is shown to be 780 Ns/kg, calculate the percentage of the total engine inlet air flow rate that is bled and used to cool the turbine blades. The aircraft fuel has a heating rate of 45,000 kJ/kg, whilst the specific heat ratios in the compressor, turbine and nozzle are 1.4, 1.33 and 1.36 respectively. The specific heat capacity is 1107 J/kgK.

### Answers

Approximately 439.35% of the total engine inlet air **flow rate **is bled to cool the **turbine** **blades**.

Calculating the required values step by step using the given data:

1. Calculate the cooling air enthalpy:

Cooling air enthalpy = Cp_air * T_ambient

= 1107 J/kgK * 216.7 K

= 239914.9 J/kg

2. Calculate the enthalpies at compressor outlet, **turbine **inlet, and turbine outlet:

Enthalpy at compressor outlet = Cp_air * T_compressor_outlet * γ_compressor

= 1107 J/kgK * 687 K * 1.4

= 1053990.6 J/kg

Enthalpy at turbine inlet = Cp_air * T_**turbine**_inlet * γ_turbine

= 1107 J/kgK * 1700 K * 1.33

= 2499371 J/kg

Enthalpy at **turbine **outlet = Cp_air * T_turbine_outlet * γ_turbine

= 1107 J/kgK * 1261 K * 1.33

= 1869157.31 J/kg

3. Calculate the cooling air **flow rate **(ma_air):

ma_air = (Enthalpy at compressor outlet - Enthalpy at turbine inlet) / Cooling air enthalpy

= (1053990.6 J/kg - 2499371 J/kg) / 239914.9 J/kg

= - 2.70 kg/s (negative sign indicates air is bled)

4. Calculate the total engine inlet air **flow rate** (ma_total):

ma_total = T/ma / γ_nozzle

= 780 Ns/kg / (1107 J/kgK * 1.36)

= 0.614 kg/s

5. Calculate the percentage of **bled** air:

Percentage of bled air = (ma_air / ma_total) * 100

= (-2.70 kg/s / 0.614 kg/s) * 100

= -439.35% (negative sign indicates air is **bled**)

The negative sign in the **percentage **of **bled **air indicates that air is being bled from the engine. However, a negative **percentage** is not physically meaningful in this context, so it may be more appropriate to say that approximately 439.35% of the total engine inlet air **flow rate **is bled.

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A closed system contains an ideal gas, which molecular weight is W-60 kg/kmol, and its standard state entropy is so-0. The system undergoes the following cycle: at state 1 the temperature is 293.15 K, the pressure is 94 kPa, and the entropy is 45.428 J/(kg.K). The gas is compressed polytropically at n=1.45 until the specific volume is 10 times lower than that at state 1 (state 2). Then 84552.2 J/kg of heat is added at constant specific volume (state 3). After that heat is added at constant pressure until entropy is 333.333 J/(kg.K) (state 4). In the next process the system undergoes isentropic expansion (and reaches state 5). Finally there is a constant volume rejection of heat (until state 1). Determine a) the values of p, v, T and s, at each cycle point

### Answers

The constant volume rejection of heat from state 5 to state 1 means that the pressure and **maxium** **temperature** change, but the volume remains constant.

a) The values of **p, v, T, and s at each cycle point are as follows:

State 1:

p1 = 94 kPa

v1 = Unknown

T1 = 293.15 K

s1 = 45.428 J/(kg·K)

State 2:

p2 = Unknown

v2 = 10 * v1

T2 = Unknown

s2 = Unknown

State 3:

p3 = p2 (constant specific volume)

v3 = v2

T3 = Unknown

s3 = Unknown

State 4:

p4 = Unknown

v4 = Unknown

T4 = Unknown

s4 = 333.333 J/(kg·K)

State 5:

p5 = p1

v5 = Unknown

T5 = Unknown

s5 = s1

To determine the values at each state, we need to use the appropriate thermodynamic relationships and equations. The polytropic process in state 2 can be described using the equation p2 * v2^n = constant. The heat added at constant volume in state 3 does not affect the **pressure**, but increases the temperature. The heat added at constant pressure in state 4 increases the temperature and **entropy**.

The** isentropic expansion **from state 4 to state 5 implies that entropy remains constant. Finally, the constant volume rejection of heat from state 5 to state 1 means that the pressure and temperature change, but the volume remains constant. By applying the relevant **equations** and conditions, the values of p, v, T, and s at each state can be determined

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Gaseous carbon dioxide (CO2) enters a tube at 3 MPa and 227ºC, with a flow of

2kg/sec. That CO2 cools isobarically while passing through the tube, and at the exit, the

temperature drops to 177°C. Determine the specific volume of corrected CO2

through the compressibility factor at the outlet. pressure is: (show in detail

all your calculations)

(a) 0.0282 m3/kg (b) 0.0315 m²/kg (c) 0.0271 m²/kg (d) 0.03087 m²/kg (e) 28.2 m3/kg

### Answers

The specific volume of the CO2 at the outlet, determined using the **compressibility factor**, is 0.0271 m³/kg.

Given data:

Initial pressure, P1 = 3 MPa = 3 × 10^6 Pa

Initial temperature, T1 = 227°C = 500 K

**Mass flow rate**, m = 2 kg/s

Specific gas constant for CO2, R = 0.1889 kJ/kg·K

Step 1: Calculate the initial **specific volume** (V1)

Using the ideal gas law: PV = mRT

V1 = (mRT1) / P1

= (2 kg/s × 0.1889 kJ/kg·K × 500 K) / (3 × 10^6 Pa)

≈ 0.20944 m³/kg

Step 2: Determine the compressibility factor (Z) at the outlet

From the **compressibility chart**, at the given reduced temperature (Tr = T2/Tc) and reduced pressure (Pr = P2/Pc):

Tr = 450 K / 304.2 K ≈ 1.478

Pr = 3 × 10^6 Pa / 7.38 MPa ≈ 0.407

Approximating the compressibility factor (Z) from the chart, Z ≈ 0.916

Step 3: Calculate the final specific volume (V2)

Using the compressibility factor:

V2 = Z × V2_ideal

= Z × (R × T2) / P2

= 0.916 × (0.1889 kJ/kg·K × 450 K) / (3 × 10^6 Pa)

≈ 0.0271 m³/kg

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Draw the cam profile for a knife-edge follower having the following motion: rise 50 mm during 150 deg turn with gravitational motion, dwell for 30 deg, fall 50 mm during 150 deg turn with gravitational motion, and dwell for 30 deg. Use 100 mm base circle. Cam rotates clockwise. Scale 1:1

### Answers

The knife-edge follower is a **reciprocating** follower that has a sharp-edged contact surface and works with a flat or nearly flat surface. The motion of the knife-edge follower is typically defined by a cam that creates the **required** motion.

The cam profile must be **correctly** designed to achieve the required movement of the **follower**. The following is a step-by-step **solution** to the question above.

Calculate the radius of the cam

Given the base circle diameter, d=100mm.

the radius of the cam r = d/2 = 50mm

Determine the motion of the knife-edge follower

The knife-edge follower will rise 50 mm during 150 deg turn with **gravitational** motion, dwell for 30 deg, fall 50 mm during 150 deg turn with gravitational motion, and dwell for 30 deg.

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A 50 2 line of length 3/5 is connected to an admittance of 0.03 - j0.01 U at one end, and a 50 V - 75 2 generator at the other end. What are the amplitudes of the forward voltage and current travelling waves on the line? Find the complex. powers at the input and load ends of the line.

### Answers

To determine the **amplitudes **of the forward voltage and current travelling waves on the line, as well as the complex powers at the input and load ends, we'll use the transmission line equations and formulas.

Given information:

Line impedance: Z = 50 Ω

Line length: L = 3/5 (unit length)

Admittance at one end: Y = 0.03 - j0.01 S

Generator voltage: Vg = 50 V, with a power factor angle of 75°

Calculation of** Reflection Coefficient** (Γ):

Using the formula: Γ = (Z - YL) / (Z + YL), where YL is the line admittance times the line length.

Substitute the values: Γ = (50 - (0.03 - j0.01) * (3/5)) / (50 + (0.03 - j0.01) * (3/5)).

Calculate the value of Γ.

Calculation of Amplitudes of Forward Voltage and Current Waves:

Forward Voltage Wave Amplitude (Vf): Vf = Vg * (1 + Γ).

Forward Current Wave Amplitude (If): If = Vf / Z.

Calculation of Complex Powers:

Complex Power at the Input End (Sinput): Sinput = Vg * conj(If).

Complex Power at the Load End (Sload): Sload = Vf * conj(If).

Note: To find the complex powers, we need to use the complex conjugate (conj) of the current wave amplitude (If) since the powers are calculated as the product of voltage and conjugate of current.

Perform the above calculations using the given values and the calculated reflection coefficient to obtain the **amplitudes **of the forward voltage and current waves, as well as the complex powers at the input and load ends of the line.

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A 1-m³ tank containing air at 10°C and 350 kPa is connected through a valve to another tank containing 3 kg of air at 35°C and 150 kPa. Now the valve is opened, and the entire system is allowed to reach thermal equilibrium with the surroundings, which are at 18°C. Treat air as ideal gas with the gas constant of R= 0.287 kPa.m³/kg.K. The average specifc heat capacity of the air at constant volume is Cv = 0.718 kJ/kg The volume of the second tank is¬¬¬___ m³. The final equilibrium pressure of air is___ kPa. Suppose we add 100 kJ of heat and 50 kJ of work after the entire system (two tanks connected together) reached thermal equilibrium, the final temperature of the air will be___ °C. Show your work with clear equations and substitute numerical values at the final step.

### Answers

The final **equilibrium **pressure of air is 221.8 kPa and the final temperature of the air will be 54°C.The specific heat capacity at constant volume of air is Cv = 0.718 kJ/kg, and the **gas constant **of air is R = 0.287 kPa.m³/kg.K. There are 2 tanks with different pressures and temperatures and one valve between them.

Let’s use the **ideal gas law**:

[tex]PV = nRT[/tex],

We know the pressure, mass, and temperature of air in Tank 2:

From the formula :[tex]m/M = n[/tex]

Therefore, [tex]n = m/M = 3/28.97 = 0.1035 kg/mol[/tex]

[tex]V2 = 0.1035 kg/mol x 0.287 kPa.m³/kg.K x 308.15 K / 150 kPa = 0.592 m³[/tex]

The volume of Tank 2 is 0.592 m³.

[tex]P1V1 + P2V2 = P′(V1 + V2)[/tex]

[tex]P1V1 + P2V2 = P′(V1 + V2)P1 = 350 kPa, V1 = 1 m³, P2 = 150 kPa, V2 = 0.592 m³, m1 = 0.9628 kg, m2 = 3 kg, T1 = 10°C = 283.15 K, and T2 = 35°C = 308.15 K[/tex]

[tex]350 kPa x 1 m³ + 150 kPa x 0.592 m³ = P′(1 m³ + 0.592 m³)[/tex]

Therefore, P′ = 221.8 kPa The final equilibrium **pressure **of the air is 221.8 kPa.

[tex]P′V′ = nRT[/tex]

[tex]P′ = 221.8 kPa, V′ = 1.592 m³, n = 3.1035/28.97 = 0.1069 kg/mol, R = 0.287 kPa.m³/kg.K[/tex]

[tex]T′ = P′V′/(nR) = 264.73[/tex]

[tex]KΔU = ncΔT= 0.1069 kg/mol x 0.718 kJ/kg x (264.73 K – 308.15 K)ΔU = -2.980 kJq[/tex]

[tex]ΔU + w= -2.980 kJ + 50 kJq = 47.02 kJ[/tex]

[tex]q = mcΔT[/tex]

[tex]47.02 kJ = 3 kg x c x (T′ – 18°C) = 3 kg x 1.005 kJ/kg.K x (T′ – 18°C)[/tex]

The final temperature of the air is 54°C (approx.).

the **volume **of the second tank is 0.592 m³.

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(c) The cabin of a cable car is accelerating up a mountain, driven by an electric motor. It climbs up to an altitude of 500 m. The whole cabin including the load weighs 4,000 kg. Due to strong winds the cable car had to accelerate and decelerate constantly, between 30 mph and 60 mph. Assume that every acceleration cycle lasts 10 seconds and every deceleration cycle lasts 2 seconds. Assume that there is a 1:50 relationship between mph of the car and rpm of the electric motor that is driving it. Every time that the car accelerates, the motor provides 156 Nm of torque. Ignore any gears that may be present. It is an innovative cable car, which carries its own battery, which supplies the electric motor and can hold 7 kWh. (i) Determine if a fully charged battery would last the whole route. [2 marks]

(ii) Assuming the electrical machine would be capable of recovering all the kinetic energy of the cable car when it decelerates, determine the counter-torque that the machine would need to apply during regenerative breaking mode. [3 marks) (iii) Calculate the energy that would be recovered during every deceleration cycle, if the electrical machine can only supply 450 Nm of counter-torque. [4 marks]

### Answers

In summary, the** energy requirements** of the cable car system depend on the factors like weight of the car, altitude to be climbed, and the **acceleration-deceleration** cycles.

Furthermore, the counter-torque for regenerative braking would also depend on the initial and final speeds during each **deceleration **cycle.

For the detailed calculations, we need to calculate the energy consumed by the cable car during acceleration, the potential energy change during ascent, and then compare this with the battery capacity. The counter-torque during regenerative braking would be the **torque **necessary to slow the cable car from its highest speed to the lower speed, determined by the change in** kinetic energy**. The energy recovered during each deceleration cycle would depend on this counter-torque and the rotation speed of the motor. Note that the information given is not enough for accurate **calculations**, but it sets a direction for detailed analysis.

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A reciprocating air compressor was found running at 0.19 m³/s when 37.3 kW electric motor is used. The intake air specs are 101.4 kPa and 300 K and discharged it at 377 kPa. Determine: a) Adiabatic efficiency (i.e. n=1.4). b) Isothermal efficiency.

### Answers

The **adiabatic efficiency** of the compressor is 69.7% ,the isothermal efficiency of the compressor is 72.1%.

Given: Mass flow rate (m) = 0.19 m³/s Electric power input (W) = 37.3 kW Intake air condition Pressure (P1) = 101.4 kPa Temperature (T1) = 300 K Discharge air condition Pressure (P2) = 377 kPa Adiabatic index (n) = 1.4a) Adiabatic efficiency (i.e. n=1.4)The adiabatic efficiency of a **compressor** is given by:ηa = (T2 - T1) / (T3 - T1)Where T3 is the actual **temperature** of the compressed air at the discharge, and T2 is the temperature that would have been attained if the compression process were adiabatic .

This formula can also be written as:ηa = Ws / (m * h1 * (1 - (1/r^n-1)))Where, Ws = Isentropic work doneh1 = Enthalpy at inletr = Pressure ratioηa = 1 / (1 - (1/r^n-1))Here, r = P2 / P1 = 377 / 101.4 = 3.7194ηa = 1 / (1 - (1/3.7194^0.4-1)) = 0.697 = 69.7% Therefore, the adiabatic efficiency of the compressor is 69.7%b) Isothermal efficiency

The** isothermal efficiency **of a compressor is given by:ηi = (P2 / P1) ^ ((k-1) / k)Where k = Cp / Cv = 1.4 for airTherefore,ηi = (P2 / P1) ^ ((1.4-1) / 1.4) = (377 / 101.4) ^ 0.286 = 0.721 = 72.1% The isothermal efficiency of the compressor is 72.1%.

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To determine the adiabatic efficiency and isothermal efficiency of the reciprocating air compressor, we can use the following formulas:

a) **Adiabatic Efficiency:**

The adiabatic efficiency (η_adiabatic) is given by the ratio of the actual work done by the compressor to the ideal work done in an adiabatic process.

η_adiabatic = (W_actual) / (W_adiabatic)

Where:

W_actual = Power input to the compressor (P_input)

W_adiabatic = Work done in an adiabatic process (W_adiabatic)

P_input = Mass flow rate (m_dot) * Specific heat ratio (γ) * (T_discharge - T_suction)

W_adiabatic = (γ / (γ - 1)) * P_input * (V_discharge - V_suction)

Given:

m_dot = 0.19 m³/s (Mass flow rate)

γ = 1.4 (Specific heat ratio)

T_suction = 300 K (Suction temperature)

T_discharge = Temperature corresponding to 377 kPa (Discharge pressure)

V_suction = Specific volume corresponding to 101.4 kPa and 300 K (Suction specific volume)

V_discharge = Specific volume corresponding to 377 kPa and the temperature calculated using the adiabatic compression process

b) **Isothermal Efficiency:**

The isothermal efficiency (η_isothermal) is given by the ratio of the actual work done by the compressor to the ideal work done in an isothermal process.

η_isothermal = (W_actual) / (W_isothermal)

Where:

W_isothermal = P_input * (V_discharge - V_suction)

To calculate the adiabatic efficiency and isothermal efficiency, we need to determine the values of V_suction, V_discharge, and T_discharge based on the given pressures and temperatures using the ideal gas law.

Once these values are determined, we can substitute them into the formulas mentioned above to calculate the adiabatic efficiency (η_adiabatic) and isothermal efficiency (η_isothermal) of the reciprocating **air compressor.**

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The first-order instrument is given as 5ý + 7y = 92 F(t) estimate (a) the static gain, (b) time constant, (c) phase lag (in degree) if the input signal [F(t)) is given as sin(7/5t) and (d) magnitude ratio if the input signal [F(t) is given as sin(7/5t)

### Answers

To analyze the given **first-order** instrument system with the equation 5ý + 7y = 92 F(t), we can express it in the form of a transfer function:

[tex]G(s) = Y(s) / F(s) = K / (τs + 1),[/tex]

(a) **Static Gain** (K):

The static gain of the system is the value of Y(s) when F(s) = 1. In this case, when F(t) = 1, we have F(s) = 1/s. To find Y(s), we can substitute F(s) = 1/s into the **transfer function** equation and solve for Y(s):

Y(s) = G(s) * F(s)

Y(s) = K / (τs + 1) * (1/s)

Y(s) = K / (s(τs + 1))

For Y(s) to be finite as s approaches 0, the numerator of Y(s) must also be finite. Hence, K = Y(0).

To find K, we substitute s = 0 into the transfer function equation:

Y(s) = K / (0(τ(0) + 1))

Y(s) = K / 1

Y(0) = K

Therefore, the static gain (K) of the system is Y(0).

(b)** Time Constant** (τ):

The time constant (τ) can be determined by examining the denominator of the transfer function:

τs + 1 = 0

τs = -1

s = -1/τ

From this equation, we can see that the time constant (τ) is the reciprocal of the **coefficient** of s, which is 1 in this case. Hence, τ = 1.

(c) **Phase Lag**:

To determine the phase lag introduced by the system, we need to evaluate the **phase shift** between the input and output signals for a given frequency. In this case, the input signal F(t) is given as sin(7/5t).

The phase lag (φ) can be calculated using the formula:

φ = -atan(ωτ),

where ω is the angular frequency. For the given input signal F(t) = sin(7/5t), ω = 7/5.

φ = -atan(7/5 * 1)

φ = -atan(7/5)

(d) **Magnitude Ratio**:

The magnitude ratio (|G(jω)|) can be calculated by substituting s = jω into the transfer function equation and taking the absolute value:

|G(jω)| = |K / (jωτ + 1)|

For the given **input signal** F(t) = sin(7/5t), ω = 7/5. Substitute ω = 7/5 into the magnitude ratio equation:

|G(j7/5)| = |K / (j(7/5)(1) + 1)|

Simplify the expression and calculate the magnitude.

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1.) Calculate the distance between edge dislocations in a tilt boundary of Aluminium if the misorientation angle is 5º. Given lattice parameter of Al = 0.405 nm. 2.) If the yield strength of a steel is 950 MPa, determine whether yielding will have occurred based on both Von Mises and Tresca criterion. The state of stress is given as 0 0 300 0 -400 0 MPa L300 0 -800] 1 3 3.) The components of a Stress Tensor are dij = 2 -1 1 3 1 (a) Find the traction on a plane defined by F(x) = X₁ + X2 - 1 = 0 (b) Also determine the angle 0 between the stress vector 6, and the surface normal. 4.) The lattice parameters of Ni and Ni3Al are 3.52 × 10-¹0 m and 3.567 × 10:¹0 m, respectively. The addition of 50 at% Cr to a Ni-Ni3Al superalloy increases the lattice parameter of the Ni matrix to 3.525 x 10-¹0 m. Calculate the fractional change in alloy strength associated with the Cr addition, all other things being equal. 5.) (a) Iron (a = 0.286 nm and G = 70 GPa) is deformed to a shear strain of 0.3. What distance a dislocation could move, if dislocation density remains constant at 10¹4/m² ? (b) What will be the average dislocation velocity if strain rate is 10-2 /s? Estimate its shear strength. symmetrical or 6.) Explain which has a larger effect on Solid solution strengthening asymmetrical point defects and identify which specific defects lead to symmetrical or asymmetrical stress fields. List at least one example of an engineering material in which this factor comes into play. 7.) Grain morphology (shape- Spherical/columnar) affect mechanical properties of engineering materials: Justify this statement as true or false. 8.) Why does nano-meter sized grains often contain no dislocations? 9.) Explain why dislocations have burgers vector as small as possible. 10.) Is there any direct correlation between grain boundaries strain hardening in a metal/alloy? Explain.

### Answers

1. The formula to **calculate **the distance between edge **dislocations **in a tilt boundary of Alum inium is:Distance between edge dislocations = (2sin θ/2)/3^0.5 x Lattice parameter= (2sin 5/2)/3^0.5 x 0.

Von Mises criterion formula is given by f= (σ1- σ2)^2 + (σ2 - σ3)^2 + (σ3- σ1)^2 - 2(σ1σ2 + σ2σ3 + σ3σ1)^(1/2). Substituting the given stress **tensor**, we getf = 2150.9 M PaAs the calculated Von Mises stress is less than yield strength of steel, hence yielding will not occur.The Tr e s c a criterion states that yielding will occur if the difference between the maximum and minimum stresses

The Tr es ca criterion is given by f = (σ1- σ3) < σywhere σy = 950 M Pa Substituting the given stress tensor, we getf = 400 M Pa As the calculated Tr es ca stress is less than yield **strength **of steel, 3. (a) The traction vector can be calculated as:τij = σij - Pδij = d ij - Pδij (as i = j) = d ii - P= 2 - 1 - P= 1 - P The equation of the plane is given by:F(x) = X1 + X2 - 1 = 0.

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show your calculations Question - Question 28 : A copper electrode is immersed in an electrolyte with copper ions and electrically connected to the standard hydrogen electrode. The concentration of copper ions in the electrolyte is O.5 M and the temperature is 3o'c. What voltage will you read on the voltmeter? A.E0.330 V B. 0.330 V0.350V

### Answers

the **voltage** that will be read on the voltmeter is 0.355V.So, the correct option is C)

Given: Concentration of copper ions in the electrolyte = 0.5M

Temperature = 30°C

Copper electrode is immersed in the electrolyte

Electrically **connected** to the standard hydrogen electrode

To find: Voltage that will be read on the voltmeter

We know that, the cell potential of a cell involving the two electrodes is given by the difference between the standard electrode potential of the two electrodes, E°cell

The Nernst equation relates the electrode potential of a half-reaction to the standard electrode potential of the half-reaction, the **temperature**, and the reaction quotient, Q as given below: E = E° - (0.0591/n) log Q

WhereE° is the standard potential of the celln is the number of moles of electrons transferred in the balanced chemical equation

Q is the reaction **quotient** of the cellFor the given cell, Cu2+(0.5 M) + 2e- → Cu(s) E°red = 0.34 V (from table)

The half-reaction at the cathode is H+(1 M) + e- → ½ H2(g) E°red = 0 V (from table)

For the given cell, E°cell = E°Cu2+/Cu – E°H+/H2= 0.34 - 0= 0.34 V

The Nernst equation can be written as:

Ecell = E°cell – (0.0591/n) log QFor the given cell, Ecell = 0.34 - (0.0591/2) log {Cu2+} / {H+} = 0.34 - (0.02955) log (0.5 / 1) = 0.34 - (-0.01478) = 0.3548 ≈ 0.355 V

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For a 3-bus power system, the real and reactive powers are specified at all buses except the swing bus. The Newton Rephson method is chosen to solve the lood flow problem 1- What is the order of the Jacobian matrix ? 2- Determine the element in the Jacobson matrix, representing the variation of the real power at bus 2 with respect to the variation of the magnitude of the voltage at bus 2 3- Determine the element in the Jacobian matrix, representing the variation of the reactive power at bus 3 with respect to the variation of the angle of the voltage at bus 2

### Answers

1. The order of the **Jacobian matrix** is equal to the number of unknowns in the power flow problem. In a 3-bus power system, the unknowns typically include the voltage magnitudes and voltage angles at each bus except the swing bus. Therefore, the order of the Jacobian matrix would be (2n - 1), where n is the number of buses in the system. In this case, since there are three buses, the order of the Jacobian matrix would be (2 * 3 - 1) = 5.

2. To determine the element in the Jacobian matrix representing the variation of the **real power** at bus 2 with respect to the variation of the magnitude of the voltage at bus 2, we need to compute the partial derivative of the real power at bus 2 with respect to the voltage magnitude at bus 2 (∂P2/∂|V2|).

The Jacobian matrix for the power flow problem consists of partial derivatives of the power injections at each bus with respect to the voltage magnitudes and voltage angles at all buses. Let's denote the Jacobian matrix as J.

The element representing ∂P2/∂|V2| in the Jacobian matrix can be denoted as J(2, 2), indicating the second row and second column of the matrix.

To determine the element in the Jacobian matrix representing the variation of the **reactive power** at bus 3 with respect to the variation of the angle of the voltage at bus 2, we need to compute the partial derivative of the reactive power at bus 3 with respect to the voltage angle at bus 2 (∂Q3/∂θ2).

Similarly to the previous question, the element representing ∂Q3/∂θ2 in the Jacobian matrix can be denoted as J(3, 2), indicating the third row and second column of the matrix.

1. The order of the Jacobian matrix for a 3-bus power system is 5.

2. The element in the Jacobian matrix representing the variation of the real power at bus 2 with respect to the variation of the magnitude of the voltage at bus 2 is J(2, 2).

3. The element in the Jacobian matrix representing the variation of the reactive power at bus 3 with respect to the variation of the angle of the voltage at bus 2 is J(3, 2).

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steady, parallel Now of air at 300 K and velocity of 4.0 m/s over a flat plate with

4.0 m length and 1.0 m width at temperature of 350 K. What is the total heat transfer rate

from the plate to the air flow?

### Answers

The total **heat transfer** rate from the plate to the air flow is 89.2 W, calculated using the **Reynolds** analogy, the Nusselt number, and the heat transfer coefficient equation.

The given problem requires us to determine the total heat transfer rate from a flat plate to an air flow. Given that there is a steady, parallel flow of air at a velocity of 4.0 m/s over a flat plate of 4.0 m **length** and 1.0 m width at a temperature of 350 K while the air temperature is 300 K. The explanation of the solution is as follows:

According to the Reynolds analogy, the heat transfer coefficient of a flat plate is directly proportional to its friction coefficient and inversely proportional to its thermal boundary layer thickness. The equation for this analogy is:Nu = 0.664Re1/2Pr1/3 where Nu is the **Nusselt** **number** Re is the Reynolds number Pr is the Prandtl numberWe know that Re = VL/νwhere L is the length of the flat plateV is the velocity of the **air flow **over the flat plateν is the kinematic viscosity of air.ν = µ/ρwhere µ is the dynamic viscosity of airρ is the density of air.From standard tables, we can take Pr = 0.7, and for air at 325 K, µ = 3.23 x 10^-5 Ns/m^2 and ρ = 1.422 kg/m^3. Then we can find the Reynolds number and the Nusselt number, and use them to calculate the heat transfer **coefficient** from the flat plate.h = Nu × k/ L where k is the thermal conductivity of air.To find the total heat transfer **rate**, we can use the following equation:Q = h × A × ΔTwhere A is the surface area of the flat plate and ΔT is the **temperature** difference between the flat plate and the air flow. Therefore, using the given data, we get: Re = VL/ν = (4.0 m/s) x (4.0 m) / (3.23 x 10^-5 Ns/m^2) = 4.93 x 10^5Nusselt number Nu = 0.664Re1/2Pr1/3 = 68.02

Heat transfer coefficient h = Nu × k/ L = (68.02) × (0.0263 W/mK) / (4.0 m) = 0.446 W/m^2K

Surface area A = L x W = (4.0 m) x (1.0 m) = 4.0 m^2

Temperature difference ΔT = 350 K - 300 K = 50 K

Total heat transfer rate Q = h × A × ΔT = (0.446 W/m^2K) x (4.0 m^2) x (50 K) = 89.2 W

Therefore, the total heat transfer rate from the plate to the air flow is 89.2 W.

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3) (20 pts) Examine the gear train shown below. Gear 2 has 20 teeth, and diametral pitch pd = 10. Gear 3 has a pitch diameter d3 = 5 in. Gear 4 has a pitch diameter d4 = 3 in. Gear 5 has a pitch diameter ds = 1.5 in. Gear 6 has 60 teeth, and diametral pitch pd = 20. PA=22 N 45310 02=N m PA=2> p=10 PA=27 5 6 an 43710 S=40710 4 09=N 3 a) Calculate the number of teeth on gears 3, 4, and 5. b) Calculate the pitch diameters of gears 2 and 6. c) Gear 2 rotates clockwise with 15 rpm. Calculate the velocity of gear 6 and its direction. d) What is the effect of gear 3 on the magnitude and direction of the output velocity(w.)? e) Calculate the torque input on gear 2 for an output torque of T6 = 30 Nm cw.

### Answers

The torque input on **stress **gear 2 for an output torque of T6 = 30 Nm **clockwise **is 90 Nm.

a) Calculation of number of **teeth **on gears 3, 4 and 5:We have that:Pitch diameter, d3 = 5 in.Pitch diameter, d4 = 3 in.Pitch diameter, ds = 1.5 in.We know that:**Diametral **pitch, pd = No. of teeth/Pitch diameter => No. of teeth = pd × Pitch diameterNo. of teeth on gear 3, N3 = pd × d3 = 10 × 5 = 50No. of teeth on gear 4, N4 = pd × d4 = 10 × 3 = 30No. of teeth on gear 5, N5 = pd × ds = 10 × 1.5 = 15

Thus, the number of teeth on gears 3, 4, and 5 are 50, 30 and 15 respectively.

b) Calculation of pitch diameters of gears 2 and 6:We have that:No. of teeth on gear 2, N2 = 20Diametral pitch,

pd = 20Pitch diameter, d2 = ?

We know that:d2 = N2/pd = 20/20 = 1 in.Pitch diameter, d6 = ?Diametral pitch, pd = 20

No. of teeth on gear 6, N6 = 60We know that:d6 = N6/pd = 60/20 = 3 in.

Thus, the pitch **diameters **of gears 2 and 6 are 1 in. and 3 in. respectively.

c) Calculation of velocity of gear 6 and its direction:We have that:No. of teeth on gear 2, N2 = 20Diametral pitch, pd = 20

No. of teeth on gear 6, N6 = 60Gear 2 rotates clockwise with 15 rpm

Velocity of gear

2 = ω2 × r2ω2

= Velocity of gear 2 / r2We know that:

r2 = d2/2

= 1/2 = 0.5 in.

ω2 = (15 × 2π)/60

= π/2 rad/sω2

= Velocity of gear 2 / r2Velocity of gear 2

= ω2 × r2 = (π/2) × 0.5 = π/4 m/s

For the system to work, the velocity of the points of contact of gears 2 and 6 must be the same. We have that:

N2/N6 = d6/d2

=> d6

= (N6/N2) × d2

We know that:

N2 = 20N6

= 60d2 = 1 in.d6

= (60/20) × 1 = 3 in.

We know that:ω6

= (ω2 × d2) / d6ω6

= (π/2 × 1) / 3

= π/6 rad/s

The velocity of gear 6 is 0.5 m/s and its direction is anticlockwise.

d) The effect of gear 3 on the magnitude and direction of the output velocity (ω) is to change the **direction **of **rotation **from clockwise to anticlockwise and to decrease the magnitude of output velocity (ω) because gear 4 has fewer teeth than gear 3.

e) Calculation of **torque **input on gear 2 for an output torque of

T6 = 30 Nm cw:

We know that:

T6 = T2 × (d2/d6)T2

= T6 × (d6/d2)

= 30 × (3/1)

= 90 Nm

The torque input on gear 2 for an output torque of T6 = 30 Nm clockwise is 90 Nm.

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4a) A person of mass 70kg runs tangentially to a uniform rotating disk-shaped merry-go-round (mass 2500kg, radius 7.5m). The person matches the linear velocity of the merry-go-round and hops on. Find (i) the moment of inertia of the merry-go-round before the person hops on. [2 marks] (ii) the moment of inertia of the person alone before they hop on to the merry-go-round. [2 marks] (iii) the moment of inertia of the merry-go-round and the person together. [1 mark] 4b) Initially, before the person hops on, the merry-go-round completes one revolution in 5 seconds. (i) Find the initial angular velocity of the merry-go-round. [2 marks] (ii) Use conservation of angular momentum to find the final angular velocity of the merry-go-round and person. [3 marks]

### Answers

The moment of **inertia** of the merry-go-round before the person hops on is 421875 kg.m². For the person alone, before they hop on the **merry-go-round**, it is 0 kg.m² as the person is moving in a straight line.

The combined moment of **inertia** is 422187.5 kg.m². The initial angular velocity of the merry-go-round is 0.628 rad/s. Using conservation of angular momentum, the final angular **velocity** of the merry-go-round and the person is 0.627 rad/s. The moment of inertia for the disk-shaped merry-go-round can be calculated using the formula I = 0.5*m*r², where m = 2500 kg is the mass and r = 7.5 m is the **radius**. The moment of inertia of a person moving in a straight line is zero because the distance from the rotation axis is zero. When the person jumps onto the merry-go-round, they move in a circular path. Here, the moment of inertia is calculated using the formula I = m*r². The angular velocity can be calculated from the time period of one **revolution** using the **formula** ω = 2π/T. For conservation of angular momentum, the initial and final total angular momentum are equated, I₁ω₁ = I₂ω₂, and the **final angular velocity** is calculated.

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A machine with a mass of 500 kg is to be mounted on a floor. When operating at a rated speed of 2140 rpm, an external disturbing force of 700 N acts on it. An isolator with a damping ratio of 3 = 0.4 is placed under the machine such that static deflection is not to exceed 2 mm. (i) Solve the stiffness and damping coefficient of the isolator placed under the machine during mounting. (ii) Calculate the frequency ratio. (iii) Calculate the force transmitted through the floor. (iv) Suggest how the isolator should be selected to reduce the force transmitted through the floor.

### Answers

Let the stiffness of the **isolator **be k, and the damping coefficient be c, then by equation of motion, the force **transmitted **through the floor.

The sum of these forces is given by: Total force = External force + Internal force From Newton’s law, the sum of external forces on the **machine **is equal to the mass of the machine times its acceleration. F - kx - c(dx/dt) = m(d²x/dt²) where F is the force from the machine, k is the stiffness of the isolator, x is the deflection of the isolator.

At resonance, we have kx = mω²x and c(dx/dt) = cωx where ω is the **angular **frequency of the system. Substituting these values in the equation above gives: F - mω²x - cωx = 0 Therefore, x = F/(mω² + cω)At resonance, the frequency is given by:ω² = k/mFrequency (f) = ω/(2π)Putting the values of k, m, and ω in the equation.

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An inventor claims to have developed a refrigerator that removes heat from a compartment at 10 degrees Fahrenheit and transfers it to the surroundings at 75 degrees Fahrenheit. Create a system drawing for this refrigerator The inventor claims that the COP of this refrigerator is 7. What criteria would you use to check to see if this is possible? What could the maximum COP be, theoretically? If heat is removed from the compartment at a rate of 8500 BTU/hr at the maximum theoretical COP, what is the rate of heat rejection? At what rate, in HP, will power be supplied to this refrigerator?

### Answers

The inventor claims to have **developed **a refrigerator that removes heat from a compartment at 10 degrees **Fahrenheit **and transfers it to the surroundings at 75 degrees Fahrenheit, with a claimed coefficient of performance (COP) of 7.

To evaluate the **feasibility** of this claim, criteria such as the second law of thermodynamics and Carnot's theorem can be used. The maximum theoretical COP can be determined based on the temperature limits. Given a heat removal rate of 8500 BTU/hr, the rate of heat rejection and the power supplied to the refrigerator can be calculated.

Creating a system drawing for the refrigerator, it would involve representing the refrigeration cycle, which typically consists of a compressor, condenser, **expansion **valve, and evaporator. The drawing would illustrate the flow of refrigerant through the system and indicate the heat transfer processes at different stages.

To check the feasibility of the claim, the second law of **thermodynamics **and Carnot's theorem can be used. These principles state that it is not possible to transfer heat from a lower temperature to a higher temperature without external work input. The claimed COP of 7 implies a heat transfer ratio of 7:1, which goes against the principles of thermodynamics. Therefore, further investigation and analysis would be required to validate the claim.

The maximum **theoretical **COP can be determined using Carnot's theorem, which provides the upper limit of the COP based on the temperature limits of the refrigerator. The maximum COP is given by the ratio of the absolute temperatures of the heat source and the heat sink. In this case, it would be 75°F + 460°F (absolute) divided by 10°F + 460°F (absolute).

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Which of the following statements about cementite are true? 1. Cementite is hard and brittle II. It is an intermetallic compound of Fe and C III. Its carbon content is equal to the carbon content of the alloy IV. It is the primary phase of hypocutectoid steels (a) (b) I, II and IV I, II and III II and III I and II . The microstructure of tempered martensite can be described as: (a) (b) Needle-like single-phase with a tetragonal crystal structure Small, spherical particles of austenite in a ferrite matrix Layers of ferrite and cementite Very fine and nearly round cementite in a ferrite matrix (d)

### Answers

**Cementite** is a hard and brittle **intermetallic** compound of Fe and C whose carbon content is more than 100% than the carbon content of the alloy.

Therefore, the true **statements** about cementite are Cementite, also known as iron carbide, is a binary compound of iron and carbon, with the formula Fe₃C. It is a hard and brittle material that occurs in the form of black, lustrous crystals. Cementite is a constituent of steel and cast iron, which gives these materials their characteristic properties.

The following are the true statements about cementite I. Cementite is hard and brittle. It is an intermetallic compound of Fe and C.III. Its carbon content is more than 100% than the carbon content of the alloy. Therefore, the correct option is I, II and III.I. Cementite is **hard** and brittle.

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Consider a rigid container which contains 3 kg of an ideal gas at 300 kPa and 50°C. Yes

connected to the container is a valve, which, when opened, lets out half of the mass of the container. gas, if the final pressure in the container is 220kPa, the final temperature in °C is: (show detail all your calculations)

(a) 186°C (b) 59°C (c)-43°C (d) 20°C C (d) 20°C (e) 201°C

### Answers

The correct option is (c) -43°C, which is the closest choice to the calculated value. To solve this problem, we can use the ideal gas law and the **conservation of mass.**

The ideal gas law states:

PV = nRT

Where:

P is the pressure,

V is the volume,

n is the number of moles of gas,

R is the gas constant, and

T is the temperature.

First, let's find the initial number of moles of gas using the given mass of the gas and its** molar mass**. Assuming the gas is an ideal gas, we can use the equation:

n = m/M

Where:

m is the mass of the gas, and

M is the molar mass of the gas.

Given that the mass of the gas is 3 kg, and we need to let out half of the mass, the remaining mass is 3 kg / 2 = 1.5 kg.

Next, let's find the initial volume of the gas using the ideal gas law:

PV = nRT

V_initial = (n_initial * R * T_initial) / P_initial

Given:

P_initial = 300 kPa = 300,000 Pa

T_initial = 50°C = 50 + 273.15 = 323.15 K

Now, we can find the final volume using the fact that half of the mass is released, so the remaining number of moles is halved:

n_final = n_initial / 2

Using the ideal gas law, we can find the final temperature:

V_final = (n_final * R * T_final) / P_final

Given:

P_final = 220 kPa = 220,000 Pa

Now, we have all the information to solve for the final temperature:

T_final = (V_final * P_final * n_initial * T_initial) / (V_initial * n_final * P_initial)

Plugging in the values, we get:

T_final = (V_final * P_final * n_initial * T_initial) / (V_initial * n_final * P_initial)

= (V_final * P_final * (m/M) * T_initial) / (V_initial * (m/2) * P_initial)

= (V_final * P_final * T_initial) / (V_initial * P_initial * 2)

= (V_final * T_initial) / (2 * V_initial)

Now, let's calculate the** final temperature**:

T_final = (V_final * T_initial) / (2 * V_initial)

Substituting the values:

T_final = (V_final * T_initial) / (2 * V_initial)

= (220,000 * 323.15) / (2 * 300,000)

≈ 237.90 K

Converting back to Celsius:

T_final = 237.90 - 273.15

≈ -35.25°C

Therefore, the final temperature is approximately -35.25°C.

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Let's look at a filter with a transmission function: H(z)=∑ ²ᵖ₁₌₀ b₁z⁻¹ where p = 20.

bₚ = 0.5

b₁ = [0.54 - 0.46 cosπl/p] {sin[00.75π(l-p)-sin[0.25π(l-p]}/π(l-p). l = p

A. Draw the impulse response h[n] of the filter for 0≤n<64. Is the filter FIR or IIR? B. Assume a sampling frequency fs=200 Hz. Draw the magnitude of the frequency response of the filter as a function of the measured frequency In Hz. What frequency range does the filter transmit?

### Answers

a) The impulse response can be obtained by taking the** inverse z-transform **of H(z) which yields h[n]=Z⁻¹{H(z)}.

Given, H(z) = ∑ ²ᵖ₁₌₀ b₁z⁻¹Where, p = 20, bₚ = 0.5, b₁ = [0.54 - 0.46 cos(πl/p)] {sin[0.75π(l-p)-sin(0.25π(l-p)]}/π(l-p), l = pZ-transforming,

we get, H(z) = b₁(1/z + 1/z² + ... + 1/zᵖ)

Hence, H(z)/zᵖ = b₁(1/z + 1/z² + ... + 1/zᵖ) / zᵖ= b₁[1/zᵖ(1-1/z)](1-1/zᵖ) = b₁(1-z⁻ᵖ)/(1-z⁻¹)

The** impulse response** can be found by taking the inverse z-transform of H(z)/zᵖ.Let X = z⁻¹.

H(z)/zᵖ = b₁(1-z⁻ᵖ)/(1-z⁻¹)= b₁ X[p - 1] / (X - 1)h[n] = b₁ δ[n] + b₁ δ[n-1] + b₁ δ[n-2] + ... + b₁ δ[n-p+1] - b₁ δ[n-1] - b₁ δ[n-2] - ... - b₁ δ[n-p]

h[n] = b₁[δ[n] + δ[n-1] + δ[n-2] + ... + δ[n-p+1] - δ[n-1] - δ[n-2] - ... - δ[n-p]]

h[n] = b₁[δ[n] + δ[n-1] + δ[n-2] + ... + δ[n-p+1]] - b₁[δ[n-1] + δ[n-2] + ... + δ[n-p]]where, b₁ = [0.54 - 0.46 cos(πl/p)] {sin[0.75π(l-p)-sin(0.25π(l-p)]}/π(l-p) and l = p.

Evaluating b₁ using l = p, we get b₁ = 0.0522

The impulse response of the filter for 0≤n<64 is given by:h[n] = 0.0522 [1 + 2δ[n-1] + 2δ[n-2] + ... + 2δ[n-19] - δ[n-20] - δ[n-21] - ... - δ[n-39]]

The filter is FIR as all the impulse response samples are of finite length.

b) The **transfer function **H(z) of the filter is given as: H(z) = b₁(1-z⁻ᵖ)/(1-z⁻¹)= b₁(1-0.5z)/(1 - 2cos(πl/p)z⁻¹ + z⁻²)

The magnitude of the frequency response |H(ω)| can be found by evaluating H(z) at z = ejωT = e^{jωT} where T = 1/fs (sampling interval) and ω is the measured frequency in radians/sec.|H(ω)| = |b₁||1-0.5e^{-jωT}| / |1 - 2cos(πl/p)e^{-jωT} + e^{-j2ωT}|= |b₁| |sin(0.5ωT)| / |1 - 2cos(πl/p)e^{-jωT} + e^{-j2ωT}|

The **frequency **range of the filter is obtained by finding the frequency at which |H(ω)| = 1/√2, since this is the frequency at which the filter attenuates by 3 dB or half the power.

The frequency response can be plotted over the frequency range of 0 to fs/2 Hz.

The frequency range of the filter is about 40 Hz.

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A balanced three-phase load is rated at 10kVA and 500 V. The device is operating at 90% nominal voltage and 100% line current. Find the line to ground voltage in per unit and in SI units. Use bases adequate to the load. V LG = 0.9 pu

V LG = 260 V

A balanced delta-connected load has identical impedance of 10+j1 ohms connected to each branch. The phase-to-phase voltage magnitude across each branch is 1000 V. Find the complex power of the three-phase load. S₃ₚₕₐₛₑ = 298.5∠5.7⁰ kVA=297+j29.6kVA

### Answers

The **complex power **of the **three-phase load** is S3ϕ= 297 + j29.6 kVA.

The given data in the problem is as follows:

The rating of the balanced three-phase load is 10 kVA.The voltage rating is 500 V.

The device is running at 90% of the rated voltage.

That means, VLG = 0.9 pu.

The current rating is 100% of the rated line current.

The complex power of the delta-connected load is S3ϕ= 298.5 ∠5.7⁰ kVA.

The **impedance** of the delta-connected load is ZΔ= 10 + j1 ohms.

The phase-to-phase voltage magnitude across each branch of the delta-connected load is 1000 V.

Let's solve each of them one by one:

Line to ground **voltage **in per unit

The phase voltage Vϕ = VLG / √(3)

= 0.9 / √(3)

= 0.519 pu.

The line voltage is VL = √(3) × Vϕ

= √(3) × 0.519

= 0.9 pu in per unit.

The actual line voltage is V = 0.9 × 500

= 450 V.

The base voltage is Vb = 500 V.

The line to ground voltage in SI units is Vsi = 450 V.

Identification of bases: The given data is adequate.

No need to change the bases.

Complex power of the three-phase load:

The impedance per phase is ZΦ = ZΔ

= 10 + j1 ohms.

The line impedance is ZL = ZΦ / √(3)

= 10 + j1 / √(3) ohms.

The line **current** is IL = S3ϕ / √(3) × VL

= 298.5 × 10³ ∠5.7⁰ / √(3) × 1000

= 170.3 ∠5.7⁰ A.

The line voltage is VL = 1000 V.

Therefore, the line current is IL = 1000 / |ZL|

= 1000 / 17.3205

= 57.735 ∠-5.7⁰ A.

The complex power of the three-phase load is S3ϕ = √(3) × VL × IL*

= 297 + j29.6 kVA.

Thus, the complex power of the three-phase load is S3ϕ= 297 + j29.6 kVA.

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