0. Background Knowledge

Transcription

1 Background Knowledge ME 200 builds on courses you have already taken, including CHEM 115, MATH 165 and 166, and PHYS 172 and 241. You should have the following material at immediate recall. 0.1 The ideal gas law Chemists often write the ideal gas law as where is the absolute pressure, is the volume, is the amount of substance in molecules (6.02x10 23 molecules/mole), is Boltzmann s constant (1.38x10-23 J/K-molecule) and is the absolute temperature. We engineers write it as or, where is the absolute pressure, is the mass density ( ), is the specific volume ( ), is the gas constant for the substance of interest, and is the absolute temperature. 0.2 The concept of internal energy (U) for molecules Molecular internal energy is made up of nuclear vibration and rotation, electronic motion, and translational motion of the entire molecule (or atom). It depends on temperature. 0.3 Systems and idealized models A system is the entity to analyze. You should also be familiar with constructing idealized models to simplify the behavior of complicated systems. 0.4 Properties at sea level At sea level, the acceleration due to gravity is 9.81 m/s 2 and atmospheric pressure is kpa (~100 kpa).

2 Hooke s Law, Ohm s Law, and the integral for Work Hooke s Law states the force required to extend or compress a spring some distance is proportional to that distance ( ), where is the spring constant and has units of N/m. Ohm s Law ( ) relates voltage (), current (), and resistance (). Power is. The integral relating force and displacement to work is. 0.6 Calculus You should know how to integrate a variable raised to a power; the difference between total and partial derivatives; double and triple integration; and the definitions of vector dot product and vector cross product. 0.7 Linear interpolation You should be familiar with linear interpolation, but as a refresher, let s say you need to evaluate the specific internal energy of air at 274 K. You re provided with data for 270 K and 280 K, but not 274 K (Table 0-1). Table 0-1. Specific internal energy of air. Temp. [K] u [kj/kg] ? For this case we need to interpolate. For interpolation, we assume that the properties vary linearly between two adjacent rows in the property table (Figure 0-1).

3 0-3 Figure 0-1. Assume properties vary linearly between adjacent data points. We now simply equate the slopes. The slope of the line between the known points (specific internal energy at 270 K and at 280 K) must be equal to the slope of the line between the unknown point (specific internal energy at 274 K) and one of the known points (specific internal energy at 270 K or at 280 K) / / / Note that we can use either known point and will calculate the same value for specific internal energy at 274 K: kj/kg.

4 Examples Integration of a variable raised to a power A force (in N) depends on displacement (in m) as, where is some number. a) What are the units of? b) Evaluate the integral K Electrical power and energy A current of 1.5 A flows through a 27 resistor for a period of 3.6 sec. a) What is the power dissipated through the resistor (assuming its temperature remains constant)? b) What energy is lost during this dissipation process (again, assuming the resistor remains at constant temperature)?

5 Gas Constant R Complete Table 0-2, where the universal gas constant is /. Table 0-2. Values of the gas constant for selected elements and compounds. Substance Chemical formula Molecular weight (kg/kmol) R (kj/kgk) Air Ammonia NH Argon Ar Carbon dioxide CO Carbon monoxide CO Helium He Hydrogen H Methane CH Nitrogen N Oxygen O Water H 2 O 18.02

6 Barometric pressure The standard barometric pressure at sea level is 760 mm of mercury. a) What is the equivalent height of mercury in inches? b) What is the equivalent height in inches of water? The density of mercury is 13.6 times that of water.