MAE135 Homework No. 1 Spring 2010

Date assigned: March 24, 2010Due date: 5:00 pm, Tuesday, April 6, 2010

Assume calorically-perfect gas with γ=1.4 for all problems.

PROBLEM 1 (5 points)

The internal energy per unit mass is e = cvT while the kinetic energy per unit mass is 1

2V 2. For

air at room temperature (290◦K), calculate the ratio of kinetic over internal energy. Plot the result(Excel or similar) for V =0 to 1000 m/s.

PROBLEM 2 (10 points)

RdRp

The expression for moving boundary work

δW = −pdV

is usually derived using the one-dimensional piston example, as shown in class and found in numer-ous thermodynamics textbooks. It is then claimed that the relation is valid for any shape boundary.Well, let us test this claim for a cylindrical-shaped boundary with initial radius R, later compressedto a radius R − dR, where dR << R. You will be dealing with work per unit depth (you can takethe depth=1 for convenience). Show how you define the work for this boundary and prove that theabove relation still holds.

PROBLEM 2 (15 points)

p1

T1�1

p2=2p1

T2�2Isothermal

orIsentropic

A gas particle is initially at pressure p1 and has a volume V1 and temperature T1. We follow thisparticle (we track the same mass) as it undergoes a process whereby the final pressure is p2 = 2p1.Consider two processes: Isothermal and Isentropic. For each process, find:(a) the final volume ratio V2/V1;(b) the final temperature ratio T2/T1;(c) the work per unit mass w = −

∫2

1pdv done on the particle. Express your answer in the form

w/(p1v1) = . . . ;(d) the entropy change (per unit mass) normalized by the gas constant, (s2 − s1)/R.

1