Problems
Heating of a Room
A room of volume 75 m3 contains air, initially at T0 = 273 K. A heater sup- plies heat at a rate of 2000 W. Since air can leave or enter the room through small gaps in windows and doors, the pressure in the room is equal to the outside pressure of 1 bar at all times. Assume that no air enters the room.
Consider air as an ideal gas with constant specific heats, and assume that all properties inside the room are homogeneous.
1. Assuming that there are no heat losses to the environment, compute mass and temperature of the air in the room as a function of time and plot the result.
2. Consider the same problem with heat losses. For this, assume that the heat losses are proportional to the difference between the temperatures inside and outside (assume Toutside = T0 = 273 K), that is Q˙ loss = αA (T0 − T ). , is the heat transfer coefficient, and A = 60 m2 is the wall surface. Plot the resulting curves.
Heat Transfer Loss in Isobaric Pipe Flow Consider a mass flow m˙ through a pipe of length L at steady state without pressure loss; the fluid enters the pipe at temperature T1. The heat exchange of the pipe section of length dx with the environment at T0 can be described by Newton’s law as δQ˙ = α (T0 − T ) dx. For the following, assume that the fluid has constant specific heat cp and ignore kinetic and potential energies.
1. Compute the temperature of the fluid, T (x).
2. Consider a control volume just around the fluid in the pipe, and compute entropy generation and work loss in that volume. Discuss your findings.
3. Consider a control volume around the pipe whose boundaries are at the environmental state and compute entropy generation and work loss. Show that the entropy generation is positive (for any choice of temperatures).
Friction Loss in Pipe Flow (Incompressible Fluid)
Consider a mass flow m˙ through an adiabatic pipe of length L at steady state with pressure loss. The fluid enters the pipe at temperature T1 and pressure p1. Due to friction, the pressure drops as δp = −βdx along the distance dx. Assume that the fluid is incompressible and has constant specific heat c (note: this means du = cdT , but not dh = cdT !). Ignore potential and kinetic energies.
1. Compute the temperature of the fluid, T (x).
2. Consider a control volume just around the fluid in the pipe, and compute entropy generation and work loss in that volume.
Friction Loss in Pipe Flow (Ideal Gas)
Consider a mass flow m˙ through a pipe of length L at steady state with pressure loss; the fluid enters the pipe at temperature T1 and pressure p1.
Due to friction, the pressure drops as δp = −βdx along the distance dx; the pipe is adiabatic. Assume that the fluid is an ideal gas with constant specific heats. Ignore potential and kinetic energies.
1. Compute the temperature of the gas, T (x).
2. Consider a control volume just around the gas in the pipe, and compute entropy generation and work loss in that volume.
3. Compare the results for a gas with those obtained for the incompressible fluid in the previous problem (discuss!).