Fundamental of Mechanical Engineering Assignment-

Part 1-

Question 1 - A bowling ball is released with a horizontal velocity of 7 m/s and negligible angular velocity and it travels down the bowling lane of 18m. Determine the velocity of its center when it reaches the end of the bowling lane. The ball is always in constant with the bowling lane during its motion. The bowling ball may be assumed to be a uniform solid sphere with a mass of 7 kg and diameter of 200mm. The static and kinetic coefficients between the ball and the lane are μ_s = 0.15 and μ_k = 0.10, respectively.

Question 2 - A crane (as shown in figure) rotates at a constant rate ω₁ about a fixed vertical axes BC. At the same time, the beam AB is being lowered at a constant rate ω in the vertical plane.

Using a sample model of uniform rod of length L and mass m for the beam, determine the reactive force and moment at the point B when the beam is horizontal.

Question 3 - The 6-kg collar shown in Figure is released from rest in the position shown. If the spring constant k = 4 kN/m and unscratched length of the spring is 150mm, how far does the collar fall from its initial position before rebounding?

Question 4 - A mercury-in-bulb thermometer is immersed into a bath of temperature T_i, and the mercury level in the stem of radius r_s rises by a finite height X_o. If the bulb has a radius r_b, and the overall heat transfer coefficient between the bulb and the fluid of the bath is U, show that the energy balance equation for the temperature of mercury in the bulb (T_b) is given by

ρCV_b(dT_b/dt) = UA_b(T_i - T_b)

where ρ is the density of mercury in the bulb, C is the specific heat, A_b is the surface area of the bulb. Assuming that the expanded mercury of the bulb (βV_bT) is equal to the change of the mercury volume in the stem (X_oA_s), demonstrate that the output variable (X_o) to the input variable (T_i) can be expressed as

[ρCV_b/UA_b](dX_o/dt) + X_o = [βV_b/A_S]T_i

where A_S is the cross section area of the hollow stem of thermometer. Using the operator D or equivalent, demonstrate that the thermometer can be expressed as a 1^st order transfer function in terms of X_o to T_i as;

(τD + 1)X_o = KT_i

Hence, show that K = βV_b/A_S, a constant and T = ρCV_b/UA_b, is the time constant of the thermometer.

Write down the general solution of the output variable, X_o. Sketch the expected behavior of the thermometer over a finite non-dimensional time internals, t/T, say from 0 to 5.

A mercury-in-bulb master thermometer is designed with a bulb radius of 1.6 mm whilst the ratio of the hollow stem to bulb radii is 0.07. If the overall heat transfer coefficient between the thermometer and the bath fluid is 800 W/m².K, show that:

(i) the time constant (T) of the thermometer is about 4 s,

(ii) the ratio of thermometer constant (K) to the volumetric expansion coefficient of mercury (β) is about 0.1.

The following properties of mercury can be used in your calculation: Density (ρ) and specific heat (C) of mercury are 13500 kg/m³ and 140 J/kg.K, respectively.

Part 2-

Question 1 - A combined cycle power plant comprises a natural gas fired, ideal gas-turbine topping cycle and a bottoming steam-generator for the steam turbine. The air inlet pressure and temperature to the gas turbine, which has a pressure compression ratio of 8, are 1 bar and 300K, respectively. The temperature of burned gases from the combustor to the turbines is 1400K and the flue gas temperature leaving the steam generator (heat exchanger) is 520K. The bottoming cycle of the power plant is an ideal reheat Rankine cycle where the steam pressure and temperature supplied to the high pressure steam turbines are 150 bar and 450 C. Additional natural gas is fired for the reheating of steam and the conditions of reheated steam supplied to low-pressure turbine stage are 30 bar and 500^oC, respectively.

(a) For the stated steady state conditions, sketch the combined cycle on a T-s diagram.

(b) Using the thermodynamic properties of air and steam from the Tables, determine;

(i) the mass flow rate of air in the gas turbine cycle if the steam generation rate is 30 kg/s,

(ii) the rate of total heat input, and

(iii) the thermal efficiency of the combined cycle.

State all assumption made in the solution.

Question 2 - A solid aluminium shaft 1.0m long and 50mm diameter is to be replaced by a tubular steel shaft of the same length and same outer diameter so that either shaft could carry the same torque and have the same angle of twist over the total length (that is having the same torsional stiffness).

Calculate the inner diameter of the tubular steel shaft.

The following properties of steel and aluminium can be used in your calculation. Steel, Gs = 84 GPa, Aluminium Ga = 28 GPa.