Modern Engineering Thermodynamics

Problems 691
A plot of this function is called the Fanno line for the flow. Plot
T out vs. _Sp/ ð _mc v Þ for air using 0 ≤ T out ≤ T in : Take T os = 300 K and
T in = 290 K. Note that _Sp/ ð _mc v Þ is double valued in T out and its
maximum value occurs at M = 1:0: Determine the two values of
T out for which _S p = 0 when T os = 300 K and T in = 290 K.
81.* Rayleigh line. An analysis of the frictionless aergonic flow of an
ideal gas with constant specific heats traveling through a constant
area duct with heat transfer at the walls can be carried out by
combining the continuity equation and the linear momentum
rate balance equation to yield the following set of equations:
p out
p in
= 1 + kM2 in
1 + kM 2 out
2
× 1 + kM2 in
T out
T in
= M out
M in
1 + kM 2 out
ðT os Þ out
= M out
ðT os Þ in
M in
2
1 + kM in
×
1 + kM out
1 + k − 1
2 M2 out
1 + k − 1
2 M2 in
and Eq. (7.37) gives s out = s in + c p ln ðT out /T in Þ− R ln ðp out /p in Þ:
For air, with (T os ) in = 100°C, p in = 0.5 MPa, M in = 0.5,
and s in = 2.2775 kJ/(kg· K), generate the following plots for
0 ≤ M out ≤ 10:
a. T out vs. s out (this plot is called the Rayleigh line).
b. _Q / _m = c p ðT os Þ out
− ðT os Þ in vs: Mout (this is the heat transfer
per unit mass to or from the air).
c. _S p / _mc p =
Q
ð sout − s in Þ/c p − _ / _m
vs: M out,
cpTw
where T w = 1 2
ð T in + T out Þ is the mean wall temperature. Note
that s out is a maximum when M out = 1.0.