UCI MAE135
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MAE135 Homework No. 1 Spring 2010 Date assigned: March 24, 2010 Due 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 = c v T while the kinetic energy per unit mass is 1 2 V 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) R dR p 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 compressed to a radius R - dR, where dR << R. You will be dealing with work per unit depth (you can take the depth=1 for convenience). Show how you define the work for this boundary and prove that the above relation still holds. PROBLEM 2 (15 points) p 1 T 1 1 p 2 =2p 1 T 2 2 Isothermal or Isentropic A gas particle is initially at pressure p 1 and has a volume V 1 and temperature T 1 . We follow this particle (we track the same mass) as it undergoes a process whereby the final pressure is p 2 =2p 1 . Consider two processes: Isothermal and Isentropic. For each process, find: (a) the final volume ratio V 2 /V 1 ; (b) the final temperature ratio T 2 /T 1 ; (c) the work per unit mass w = - 2 1 pdv done on the particle. Express your answer in the form w/(p 1 v 1 )= ... ; (d) the entropy change (per unit mass) normalized by the gas constant, (s 2 - s 1 )/R. 1
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UCI MAE135 solution hw1
Transcript of UCI MAE135
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.
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