A fluid is confined in a cylinder by a spring-loaded, frictionless piston so that the pressure in the fluid is a linear function of the volume (p = a + bV). The internal energy of the fluid is given by the following equation
U = 34 + p V
Where U is in kJ, p in kPa, and V in m3. If the fluid changes from an initial state of 170 kPa, \(0.03{m^3}\) to a final state of 400 kPa, \(0.06{m^3}\), with no work other than that done on the piston, find the direction and magnitude of the work and heat transfer.
(AMIE, Summer 2023)
\(\begin{array}{l}{U_2} - {U_1} = 5({p_2}{V_2} - {p_1}{V_1})\\ = 5(400x0.06 - 170x0.03)\\ = 94.5\,kJ\end{array}\)
Now,
p = a + bV
170 = a + b x 0.03
400 = a + b x 0.06
Solving the above two equations, we get
a = -60
b = 7666.67
Therefore
\(\begin{array}{l}W = \int_1^2 {pdV = \int_1^2 {(a + bV)dV} } \\ = a({V_2} - {V_1}) + b\frac{{{V_2}^2 - {V_1}^2}}{2}\\ = \left( {{V_2} - {V_1}} \right)\left[ {a + \frac{b}{2}({V_1} + {V_2})} \right]\\ = (0.06 - 0.03)\left[ { - 60 + \frac{{7666.67}}{2}(0.06 + 0.03)} \right]\end{array}\)
= 8.55 J
\(\begin{array}{l}Q = W + {U_2} - {U_1}\\ = 8.55 + 94.5 = 103.05\,kJ\end{array}\)
A
turbine is supplied with steam at a gauge pressure of 1.4 MPa The
steam, after expansion in the turbine, flows into a condenser maintained
at a vacuum of 710 mm of Hg. The barometric pressure is 772 mm of Hg.
Express the inlet and exhaust steam pressure in pascal (absolute). Take
density of mercury as \(13600\,kg/{m^3}\) and acceleration due to
gravity as \(9.81\,m/{s^2}\).
(AMIE Summer 2023, 10 marks)
Given data
Gauge pressure = 1.4 MPa = \(1.4x{10^6}\,Pa\)
Vacuum pressure = 710 mm of Hg ;
Barometric pressure = 772 mm of Hg;
p = \(13600\,kg/{m^3}\)
g = \(9.81\,m/{s^2}\)
Atmospheric pressure
\(\rho gh = 13600x9.81x0.772 = 103x{10^3}\,N/{m^2}\)
Inlet steam pressure = Gauge pressure + Atmospheric pressure
\(1.4x{10^6} + 103x{10^3} = 1.503x{10^6}\,N/{m^2}\,or\,Pa\)
= 1.503 MPa
Condenser pressure = Barometric pressure - Vacuum pressure
= 772 - 710 = 62 mm of Hg = 62 x 133.3 = 8265 \(N/{m^2}\) or Pa
= 8.265 kPa
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