### Thermal Science & Engineering (Solved Numerical Problems)

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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