GATE ME solved Questions on First Law of Thermodynamics
Question #1
Steam at an initial enthalpy of 100 kJ/kg and inlet velocity of 100 m/s, enters an insulated horizontal nozzle. It leaves the nozzle at 200 m/s. The exit enthalpy (in kJ/kg) is __________
[GATE 2016 Set 2, 2 marks]
Solution
Correct Answer: 85 kJ/kg
Applying Steady flow energy equation between inlet and exit sections,
## Don’t forget to divide the V^{2 }/ 2 term by 1000 to convert it to kJ\kg
$h_{2}=85 kJ/kg$ ……(Answer)
Question #2
Work is done on an adiabatic system due to which its velocity changes from 10 m/s to 20 m/s, elevation increases by 20 m and temperature increases by 1 K. The mass of the system is 10 kg, C_{v} = 100 J/(kg.K) and gravitational acceleration is 10 m/s^{2}. If there is no change in any other component of the energy of the system, the magnitude of total work done (in kJ) on the system is _______
[GATE 2015 Set 2, 2 marks]
Solution
Correct Answer: 4.5 kJ
Given, $V_{1}=10m/s$, $V_{2}=20m/s$
Increase in elevation,
$z_{2}-z_{1}=20m$
Increase in temperature,
$T_{2}-T_{1}=1 K$
$C_{v}=100 J/kg.K$
$m=10 kg$
$Q=0$(As the process is adiabatic)
Applying Steady flow energy equation between initial and final state,
We know that $h=u+pv$ and $u=C_pT$, assuming pv=constant, we can write, $h=C_pT$
Therefore,
$W=\dot m \left [C_{v}(T_{1}-T_{2})+\frac{c_{1}^2-c_{2}^2}{2000}+\frac{(z_{1}-z_{2})}{1000} \right]$
A well-insulated rigid container of volume 1 m^{3} contains 1.0 kg of an ideal gas [C_{p} = 1000 J/(kg.K) and C_{v} = 800 J/(kg.K)] at a pressure of 105 Pa. A stirrer is rotated at constant rpm in the container for 1000 rotations and the applied torque is 100 N-m. The final temperature of the gas (in K) is
[GATE 2015 Set 1, 2 marks]
(A) 500.0
(B) 773.0
(C) 785.4
(D) 1285.4
Solution
Correct Option: (D)
According to First law of thermodynamics,
$Q=\Delta U +W…….(i)$
where,
$Q$=Heat Transfer
$\Delta U$=change in internal energy
$W$=work done
Here, $Q=0$(as the container is insulated)
$R=C_{p}-C_{v}$(R is the characteristic gas constant)
Work done= $T\times 2\pi N$(Here N is number of Ratations only NOT rpm)
$W=100\times 2\times 3.14\times 1000$
work is done on the system, therefore work done will be (-)
$W=-628kJ…….(iv)$
From eqaution (i),
$\Delta U= – W$
$U_{2}-U_{1}=-W$
$C_{v}(T_{2}-T_{1})=628$
$0.8(T_{2}-500)=628$
$T_{2}=1285K$ …..(Answer)
Question #4
Specific enthalpy and velocity of steam at an exit of steam turbine, running under steady state, are as given below:
Specific enthalpy(kJ/kg)
Velocity(m/s)
Inlet steam condition
3250
180
Outlet steam condition
2360
5
The rate of heat loss from the turbine per kg of steam flow rate is 5 kW. neglecting changes in potential energy of steam, the power developed in kW by the steam turbine per kg of steam flow rate is
[GATE 2013, 2 marks]
(a)901.2
(b)911.2
(c)17072.5
(d)17082.5
Solution
Correct Option: (A)
Given,
$h_{1}=3250 kJ/kg, h_{2}=2360 kJ/kg$
$c_{1}=180 m/s, c_{2}=5 m/s$
$\frac{dQ}{dt}=-5kW$
Applying Steady flow energy equation between initial and final state,
Air enters an adiabatic nozzle at 300 kPa, 500 K with a velocity of 10 m/s. It leaves the nozzle at 100 kPa with a velocity of 180 m/s. The inlet area is 80 cm_{2}. The specific heat of air Cp is 1008 J/kg.K.
Question #5
The exit temperature of the air is
(A) 516 K
(B) 532 K
(C) 484 K
(D) 468 K
[GATE 2012, 2 Marks]
Solution
Answer: (C)
Applying steady flow energy equation between entry and exit points,
The temperature and pressure of air in a large reservoir are 400 K and 3 bar respectively. A converging-diverging nozzle of exit area 0.005 m^{2 }is fitted to the wall of the reservoir as shown in the figure. The static pressure of air at the exit section of isentropic flow through the nozzle is 50 kPa.The characteristic gas constant and the ratio of specific heats of air are 0.287kJ/kgK and 1.4 respectively
Question #7
The density of air in kg/m^{3 }at the nozzle exit is
(A) 0.560
(B) 0.600
(C) 0727
(D) 0.800
[GATE 2011, 2 marks]
Solution
Answer: (C)
It is given that, the flow is isentropic. Hence we can use the relation,
The velocity at exit can be calculated using the relation,
$V_2=\sqrt{2C_p(T_1-T_2)}$
$V_2=\sqrt{2\times 1005\times (400-239.73)}$
$V_2=567.57m/s$ …..(ii)
To calculate the density of air at the nozzle exit, we can use the relation,
$P_2=\rho _2RT_2$
$50\times 1000=\rho _2\times 287\times 239.73 $
$\rho _2 = 0.726kg/m^3$ ……(Answer)
Question #8
The mass flow rate of air through the nozzle in kg/s is
(A) 1.30
(B) 1.77
(C) 1.85
(D) 2.06
[GATE 2011, 2 marks]
Solution
Answer: (D)
mass flow rate at nozzle exit is given by,
$\dot{m}=\rho _2A_2V_2$
$\dot{m}=0.726\times 0.005\times 567.57$
$\dot{m}=2.06kg/s$ ……(Answer)
Question #9
A balloon containing an ideal gas is initially kept in an evacuated and insulated room. The balloon ruptures and the gas fills up the entire room. Which of the following statements is TRUE at the end of the above process.
(A) The internal energy of the gas decreases from its initial value, but the enthalpy remains same
(B) The internal energy of the gas increases from its initial value, but the enthalpy remains same
(C) Both internal energy and enthalpy of the gas remain constant
(D) Both internal energy and enthalpy of the gas increase
[GATE 2008, 2 marks]
Solution
Answer: (C)
It is given that the room is insulated, Hence there will be no Heat transfer(i.e. $\delta Q=0$)
It is basically a Free expansion process, Hence the work done will also be zero(i.e. $\delta W=0$)
from, $\delta Q=dU+\delta W$, we can conclude that change in internal energy is also zero (dU=0) in this case.
Internal energy and enthalpy(h=u+pv) both are constant in this process.
Therefore, Option (C) is correct
Question #10
A rigid, insulated tank is initially evacuated. The tank is connected with a supply line through which air (assumed to ideal gas with constant specific heats) passes at 1 Mpa, 350 ^{o}C. A valve connected with the supply line is opened and the tank is charged with air until the final temperature inside the tank reaches 1 MPa.The final temperature inside the tank.
(A)is greater than 350^{o}C
(B) is less than 350 ^{o}C
(C) is equal to 350 ^{o}C
(D) may be greater than less than, or equal to 350^{o}C, depending on the volume of the tank
cannot understand the solution of problem 2
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