First Law of Thermodynamics

Abstract

The first law of thermodynamics states that the total energy of a system remains constant, even if it is converted from one form to another.

Problems

Problem 5.1

For circulation of air in a large room, an 8 hp fan is used. If we assume the room is fully insulated, then determine the internal energy increase after 1 h of fan operation.

Problem 5.2

If your mass body is 60 kg and you consume a 900-kcalorie hot fudge with whipped cream, then how much work should you do in order to run up several flights of stairs in order to burn off all those calories?

Problem 5.3

A rigid volume contains 6 ft³ of steam originally at a pressure of 400 psi and a temperature of 900 °F. Estimate the final temperature if 800 Btu of heat is added.

Problem 5.4

A frictionless piston is used to provide a constant pressure of 400 kPa in a cylinder containing steam originally at 200 °C with a volume of 2 m³. Calculate the final temperature if 3500 kJ of heat is added.

Problem 5.5

A mixture of 25% nitrogen and 75% hydrogen by volume is compressed isentropically from 300 K and 100 kPa to 500 kPa in the first stage of a multistage compressor in a fertilizer plant. The compression to still higher pressure is achieved in subsequent stages after the gas mixture is passed through intercoolers. It is desired to predict the temperature of the gas mixture after compression as well as the work required per unit mass of the mixture. Also, evaluate the entropy change for each gas. Assume that the mixture behaves like an ideal gas.

Problem 5.6

A rigid and insulated tank of volume 2 m³ contains 50% helium (by volume) at 100 kPa and 300 K. The tank is connected to high-pressure line carrying helium at 4 MPa and 600 K. The valve is opened, and the helium enters the tank until the pressure inside the tank is 4 MPa. Then, the valve is closed and the tank is isolated. Using Fig. 5.26 below, determine:

Fig. 5.26

Schematic for Problem 5.6

• (a) The final temperature of the gas mixture in the tank

• (b) The composition of the mixture in the tank

• (c) The amount of helium that enters the tank

Problem 5.7

In a certain steam plant, the turbine develops 1000 kW. The heat supplied to the steam in the boiler is 2800 kJ/kg, the heat rejected by the steam to the cooling water in the condenser is 2100 kJ/kg, and the feed-pump work required to pump the condensate back into the boiler is 5 kW. Using Fig. 5.27, calculate the steam flow rate.

Fig. 5.27

Schematic for Problem 5.7

Problem 5.8

Using the Fig. 5.28 below, determine the maximum pressure increase across 10 hp. The inlet velocity of the water is 30 ft/s (see Fig. 5.28).

Fig. 5.28

Schematic for Problem 5.8

Problem 5.9

In turbine of a gas turbine unit, the gases flow through the turbine at 17 kg/s and the power developed by turbine is 14,000 kW. The specific enthalpies of the gases at inlet and outlet are 1200 kJ/kg and 360 kJ/kg, respectively, and the velocities of the gases at inlet and outlet are 60 m/s and 150 m/s, respectively. Calculate the rate at which heat is rejected from the turbine. Find also the area of the inlet pipe given that the specific volume of the gases at inlet is 0.5 m³/kg (Fig. 5.29).

Fig. 5.29

Schematic for Problem 5.9

Problem 5.10

Air flows steadily at the rate of 0.4 kg/s through an air compressor, entering at 6 m/s with a pressure of 1 bar and a specific volume of 0.85 m³/kg and leaving at 4.5 m/s with a pressure of 6.9 bar and a specific volume of 0.16 m³/kg. The specific internal energy of the air leaving is 88 kJ/kg greater than that of the air entering. Cooling water in a jacket surrounding the cylinder absorbs heat from the air at the rate of 59 kW. Calculate the power required to drive the compressor and the inlet and outlet pipe cross-sectional areas (Fig. 5.30).

Fig. 5.30

Schematic for Problem 5.10

Problem 5.11

A steam receives a steam flow of 1.3 kg/s and the power output is 500 kW. The heat loss from the casing is negligible. Calculate:

• (i) The change of specific enthalpy across the turbine when the velocities at entrance and exit and the difference in elevation are negligible

• (ii) The change of specific enthalpy across the turbine when the velocity at entrance is 60 m/s, the velocity at exit is 360 m/s, and the inlet pipe is 3 m above the exhaust pipe

Problem 5.12

A steady flow of steam enters a condenser with a specific enthalpy of 2300 kJ/kg and a velocity of 350 m/s. The condensate leaves the condenser with a specific enthalpy of 160 kJ/kg and a velocity of 70 m/s. Calculate the heat transfer to the cooling fluid per kilogram of steam condensed.

Problem 5.13

A turbine in a steam power plant operating under steady state conditions receives 1 kg/s. of superheated steam at 3 MPa and 350 °C with a velocity of 50 m/s at an elevation of 2 m above the ground level. The steam leaves the turbine at 1o kPa with a quality of 0.95 at an elevation of 5 m above the ground level. The exit velocity of the steam is 120 m/s. The energy losses as heat from the turbine are estimated at 5 kJ/s. Using Fig. 6.6, calculate (Fig. 5.31):

Fig. 5.31

Schematic for Problem 5.13

• (i) The power output of the turbine.

• (ii) How much error will be introduced if the kinetic energy and the potential energy terms are ignored?

Problem 5.14

Consider the steady state operation of a compressor. The fluid enters the compressor at P₁ and υ₁ then leaves the compressor at P₂ and υ₂. Show that the work done on the compressor is given by $W=-{\int}_{\upsilon_1}^{\upsilon_2}\upsilon dP$. Sketch the P − υ diagram to represent the work done on the compressor and compare it with the work done if the compression is carried out in a piston-cylinder assembly under non-flow conditions.

Problem 5.15

In a steam power plant, saturated liquid water at 10 kPa enters a feed pump at the rate of 1 kg/s. The feed pump delivers the water to the boiler at a pressure of 3 MPa. Assuming that the pump is adiabatic, estimate the power input to the pump.

Problem 5.16

In an air-conditioning plant, saturated Freon-12 at −20 °C with a quality of 0.8 enters an adiabatic compressor and leaves as saturated at 40 °C. If the flow rate of Freon through the compressor is 1 kg/s, estimate the power input to the compressor.

Problem 5.17

A turbine operating under steady-flow conditions receive steam at the following state: pressure, 13.8 bar; specific volume, 0.143 m³/kg; specific internal energy, 2590 kJ/kg; and velocity, 30 m/s. The state of the steam leaving internal energy 2360 kJ/kg, velocity 90 m/s. Heat is rejected to the surrounding at the rate of 0.25 kW and the rate of steam flow through the turbine is 0.38 kg/s. Calculate the power developed by the turbine.

Problem 5.18

A nozzle is a device for increasing the velocity of a steady flowing fluid. At the inlet to a certain nozzle, the specific enthalpy of the fluid is 3025 kJ/kg and the velocity is 60 m/s. At the exit from the nozzle, the specific enthalpy is 2790 kJ/kg. The nozzle is horizontal and there is a negligible heat loss from it. Calculate:

• (i) The velocity of the fluid at exit

• (ii) The rate of flow of fluid when the inlet area is 0.1 m² and the specific volume at inlet is 0.19 m³/kg

• (iii) The exit area of the nozzle when the specific volume at the nozzle exit is 0.5 m³/kg

Problem 5.19

Steam enters an adiabatic nozzle operating at steady state at 4 bar and 200 °C with negligible velocity and exits at 2 bar with a velocity of 300 m/s. Determine the temperature of the steam leaving the nozzle.