Elementary Partial Differential Equations

Heat flow in a circular cylinder.

Consider a strand of heat-conducting material, homoge- neous with heat capacity c, thermal conductivity κ, and surface heat transfer coefficient μ. The strand is a right circular cylinder of radius R and height H. Unless otherwise indicated below, assume that no heat is being generated or destroyed inside the strand. For each of the following scenarios, set up the IBVP for the temperature distribution u in the strand. In each case, reduce the spatial dimension of the problem as far as possible, identify the independent variables (time t and a subset of the cylindrical coordinates r, θ, z), write all equations explicitly in terms of those variables, and indicate where exactly the equations are to hold. Also, whenever possible, set up the corresponding steady-state problem. If it reduces to an ODE, solve the steady-state problem and graph the solution.

(a) While the lateral surface and the top of the cylinder are perfectly insulated, the bottom is maintained at a constant temperature Tbot. The initial temperature distribution is a function f(z).(b) Same as in (a), except that the temperature at the bottom changes, at a constant rate and over a period of τ units of time, from an initial constant temperature T0 to a final constant temperature T∞, and is maintained at that final value ever after.
(c) Same as in (a), except that heat is exchanged across the top end, according to Newton’s law of cooling, with an external medium at constant temperature Text.
(d) Same as in (c), except that Newton’s law applies also on the lateral surface of the cylinder. (e) Same as in (d), except that top and bottom are perfectly insulated and the initial temperature
distribution is a function f(r).

(f) Same as in (e), except that the initial temperature distribution is a function f(r,θ) and that heat is generated inside the cylinder (for example, via Joule heating) at a constant rate G (heat units per unit time and unit volume).
(g) Same as in (f), except that heat is added also through the bottom of the cylinder, at a constant rate Q (heat units per unit time and unit area).

Explain different types of welding?

A closed, rigid tank fitted with a paddle wheel contains 2.2 kg of air, initially at 200oC, 1 bar. During an interval of 20 minutes, the paddle wheel transfers energy to the air at a rate of 1 kW. During this time interval, the air also receives energy by heat transfer at a rate of 0.5 kW. These are the only energy transfers. Assume the ideal gas model for the air, and no overall changes in kinetic or potential energy. Do not assume specific heats are constant. Determine the change in specific internal energy for the air, in kJ/kg, and the final temperature of the air, in oC. The answer to the change in internal energy is 818.2 kJ/kg. find the final temperature of the air, in ^oC.

Steam at 50 Bar and 500oC is expanded isentropically through a single stage turbine to a condenser operating at 1 bar.

Assuming the steam at turbine exit is Dry Saturated Steam and the turbine is required to produce a power output of 5.33MW. Calculate the required steam mass flow rate in kg/s to 2 decimal places.

1.      True or false: heating air de-humidifies it.

2.      What capacity (power) electric heater would be needed to raise the temperature of 300 cubic feet of concrete from 65° F to 150° F in 120 minutes? As this is electrical power, state response in watts or kilowatts.

3.      What would be the total cost of #2 fuel oil to operate a boiler expected to perform the task in question 2 if the boiler were operating at full fire with an efficiency of 80% and the cost of #2 fuel oil is $2.50/gallon?

4.      How does a chilled beam save energy in the operation of an HVAC system?

5.      What are the four major components of a vapor-compression refrigeration machine?

6.      What is the purpose of a VFD in an HVAC system?

1. How much oil (in tonnes) would be required in an oil-fired power plant to generate the equivalent amount of electrical energy as the hydropower generates in a year? The energy content of oil is 11,630 kWh/tonne and chemical to electrical energy conversion efficiency is about 30%.

2. How much (in tonnes) would be the CO2 emissions if the Oil power plant was used? Assume the oil power plant produces 0.27 kg CO2 per kWh

‘A walk in thermodynamics’: starting with a subject of your choice in Wikipedia, try and follow some links to get to a subject we have studied in thermodynamics. Write down which topics you stepped through on your journey. Try and take some time to read some of the material you come across. Do this for a couple of topics that interest you. [20]

For example, my background is in combustion so I followed this path: Combustion -> Heat of Combustion -> Enthalpy of Formation -> Enthalpy

A gasoline engine is modified to run on hydrogen. Assuming hydrogen releases heat at the rate of 120 MJ/Kg when burnt, a. Write the chemical reaction involved in burning of hydrogen in air. b. Determine ratio of mass of air to fuel (A/F ratio) c. What is the fuel consumption rate, if engine produces 100 kW? The engine operates between combustion temperature of 585ºC and exhaust temperature of 50º C? d. What is the mass flow rate of water in exhaust pipe?

A cylinder of an unknown length made of copper is used being used to support a roof is deformed elastically in tension by a force. The diameter of the cylinder is 25 mm. The reduction in diameter is 3*10^-3 mm Given the following information, what is the Stress experienced by the cylinder?

modulus of elasticity for copper:110 GPA

Poissons ratio for copper:0.34

What are the air-standard assumptions and the cold air-standard assumptions in analysing engine cycles? How an actual combustion process is being simplified in ideal engine cycles?

Suggest and briefly explain one practical method to significantly increase the production rate of a Die Casting process without the need to buy additional capital equipment.

An 8-m-long, uninsulated square duct of cross section 0.2 m X 0.2 m and relative roughness 10-3 passes through the attic space of a house. Hot air enters the duct at 1 atm and 91oC at a volume flow rate of 0.15 m3/s. The duct surface is nearly isothermal at 60oC. Determine the rate of heat loss from the duct to the attic space and the pressure difference between the inlet and outlet sections of the duct. Evaluate air properties at a bulk mean temperature of 80oC. Is this a good assumption? Taking a bulk mean fluid temperature of 80oC based on the problem statement (this assumes that the air does not loose much heat to the attic), the properties of air are: cp = 1008 J/kg∙oC k = 0.02953 W/m∙oC ν = 2.097 × 10-5 m2/s ρ = 0.9994 kg/m3 Pr = 0.7154

A single stage simple vapr compression refrigeration cycle using R12 refrigerant is operating at a condenser temperature of 40 degrees C and an evaporator temperature of -5 degrees C. If the compressor is a reciprocating type compressor with 4 cylinder, rotating at 1800 RPM, has a cylinder diameter of 5cm, stroke length to diameter ratio of 1.4, and clearance ratio of 5%. Assume a polytropic index to be 1.13. 1- Sketch the cycle flow diagram, and identify the states on the P-H diagram. Calculate the refrigeration effect, refrigeration capacity in TR, the compressor power in HP, the cycle COP, the temperature of the refrigerant after exiting the compressor, and the cycle refrigeration efficiency.

A closed piston-cylinder system undergoes a cycle:
​Process 1 to 2 is an expansion process where heat is added. The temperature remains constant.
Process 2 to 3 is a constant volume heat addition process.
​Process 3 to 4 is a compression process where heat is lost. The pressure remains constant.
​Process 4 to 1 is polytropic expansion, Pvⁿ​=constant with n=1.4 and it is adiabatic (Q=0).
​Assume air, ideal gas, with constant specific heats, k=1.4. R=.287kJ/kgK
​State 1: T=300K, P=150kPa
​State 2: T=300K, P=100kPa
State 3: P=600kPa
​State 4: P=600kPa

​a.) Find the temperature for State 3 and State 4.
​b.) For all 4 States, find volume(m³/kg)
​c.) For each process, (1-2, 2-3, 3-4, 4-1), find Q/m(kJ/kg), W/m(kJ/kg), and Δu(kJ/kg)
​d.) Plot all 4 States on a P-v diagram
​e.) Is this a power or refrigeration cycle?
​f.) Calculate the efficiency or coefficient of performance based on your answer to part e.