SO lost with this. Please help! Thank you so much. I really just don\'t even kno

Question

SO lost with this. Please help! Thank you so much. I really just don't even know where to start.

You are working for a venture capital firm and are tasked with determining if an invention should be considered for financial feasibility studies. The invention consists of a tube into which compressed air is admitted. The tube separates the compressed air at an intermediate temperature into two air flows, one hot and one cold. A schematic of the invention is shown below in fig. 1. The compressed air admitted into the device is divided such that 60% of it exits at the cold outlet and 40% at the hot outlet. The operating parameters for the device as given by the designer are given in table 1. Note that the device has two modes of operation. The compressed air in the device performs no external mechanical work and no significant heat transfer occurs within the device. Operating Parameters for Device. The mass flow rate is not required for the initial analysis For each mode of operation (Mode 1 or Mode 2) determine if the device violates the law of conservation of energy. Ignore any changes in macroscopic kinetic or potential energy in the flow. Derive all necessary equations based on the assumption that air behaves as an ideal gas withy=1.4 For a closed system consisting of n moles of an ideal gas, show that AS for a reversible process that takes the gas from state 1( n, Pi, Vi, Ti) to state 2( n, Pf, Vf, Tf) is: Your derivation should start from the 1st law of thermodynamics, the Claussius definition for change of Entropy (dS=dQrev/T) and the ideal gas equations: PV=nRT =(y-l)U. For each mode of operation (Mode 1 or Mode 2) determine if the device violates the 2nd Law of Thermodynamics. (Assume air behaves as an ideal gas with y=1.4) Comment on the feasibility of the device and whether you would recommend your company to start considering financial feasibility

Explanation / Answer

a)

Q - W = U2 - U1

We have Q = 0 and W = 0.

So U2 - U1 = 0

For cold gas, U2 - U1 = (0.6*m)*Cv*(Tc - Ti)

For hot gas, U2 - U1 = (0.4*m)*Cv*(Th - Ti)

Adding them, total U2 - U1 = m*Cv*[0.6*(Tc - Ti) + 0.4*(Th - Ti)]

Equating it to zero, m*Cv*[0.6*(Tc - Ti) + 0.4*(Th - Ti)] = 0

or 0.6*(Tc - Ti) + 0.4*(Th - Ti) = 0

Putting values for mode 1, 0.6*(-18 - 20) + 0.4*(70-20) = -2.8 which is non-zero.

Hence, mode 1 violates law of conservation of energy.

Putting values for mode 2, 0.6*(-18 - 20) + 0.4*(70-20) = -2.8 which is non-zero.

Hence, mode 2 violates law of conservation of energy.

b)

Q - W = du

Now, Q = Tds and W = p*dV

Hence, T ds = du + p*dV

Also, h = u + pv

dh = du + p*dv + v*dp

So, T ds = dh - v dp

ds = (dh / T) - (v/T)*dp

But dh / dT = Cp and pV = RT

So, ds = (Cp*dT / T) - (R/p) dp

Integration gives, s2 - s1 = Cp ln (T2 / T1) - R ln (p2 / p1)

Now, Cp = gamma*R / (gamma - 1)

So, s2 - s1 = (gamma*R / (gamma - 1)) ln (T2 / T1) - R ln (p2 / p1)

s2 - s1 = R*[(gamma / (gamma - 1)) * ln (T_f / T_i) - ln (P_f / P_i)]

For n moles,

S2 - S1 = nR*[(gamma / (gamma - 1)) * ln (T_f / T_i) - ln (P_f / P_i)]

c)

For mode 1:

For cold gas:

S2 - S1 = 0.6*8.314*[(1.4 / (1.4 - 1)) * ln ((-18 + 273) / (20+273) - ln (1 / 5)]

S2 - S1 = 6.603 J/K

For hot gas:

S2 - S1 = 0.4*8.314*[(1.4 / (1.4 - 1)) * ln ((70 + 273) / (20+273) - ln (1 / 5)]

S2 - S1 = 7.186 J/K

Net S2 - S1 = 7.186 + 6.603 = 13.789 J/K

Since S2 - S1 > 0, it does not violate 2nd law.

For mode 2:

For cold gas:

S2 - S1 = 0.6*8.314*[(1.4 / (1.4 - 1)) * ln ((-18 + 273) / (20+273) - ln (5 / 5)]

S2 - S1 = -2.425 J/K

For hot gas:

S2 - S1 = 0.4*8.314*[(1.4 / (1.4 - 1)) * ln ((70 + 273) / (20+273) - ln (5 / 5)]

S2 - S1 = 1.833 J/K

Net S2 - S1 =1.833 - 2.425 = - 0.59 J/K

Since S2 - S1 < 0, it violates 2nd law.

d)

No feasibility since it violates 1st law.

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