Figure \(P 4.67\) illustrates a system and fixed control volume at time \(t\) and the system at a short time \(\delta t\) later. The system temperature is \(T=100^{\circ} \mathrm{F}\) at time \(t\) and \(T=103^{\circ} \mathrm{F}\) at time \(t+\delta t,\) where \(\delta t=0.1 \mathrm{s} .\) The system mass, \(m,\) is 2.0 slugs, and 10 percent of it moves out of the control volume in \(\delta t=0.1\) s. The energy per unit mass \(\tilde{u}\) is \(c_{v} T,\) where \(c_{v}=32.0\) Btu/slug \(\cdot^{\circ} \mathrm{F}\). The energy \(U\) of the system at any time \(t\) is \(m \tilde{u}\). Use the system or Lagrangian approach to evaluate \(D U / D t\) of the system. Compare this result with the \(D U / D t\) evaluated with that of the material time derivative and flux terms.

Step by step solution

01

Calculate initial energy of the system

Determine the initial energy of the system at time t. Use the equation \(U = m \tilde{u}\), where \(\tilde{u}\) is the energy per unit mass which is equal to \(c_v T\). Here \(m = 2.0 slugs\), \(c_v = 32.0 Btu/slug^{°F}\), and \(T = 100^{°F}\). So the initial energy of the system, \(U_1 = m \cdot c_v \cdot T = 2.0 \cdot 32.0 \cdot 100 = 6400 Btu\).

02

Calculate final energy of the system

Determine the final energy of the system at time \(t + \delta t\), assuming that 10 percent of the mass leaves the system but the entire mass experiences the increase in temperature. Thus, the new mass \(m' = 0.9 \cdot m = 0.9 \cdot 2.0 = 1.8 slugs\), and the new temperature \(T' = 103^{°F}\). Using the same equation \(U = m \tilde{u}\), we find the final energy of the system, \(U_2 = m' \cdot c_v \cdot T' = 1.8 \cdot 32.0 \cdot 103 = 5929.6 Btu\).

03

Use the Lagrangian approach to evaluate DU/Dt

The Lagrangian system approach calculates the rate of change of energy in the system by comparing the initial and final energies at two moments in time. Therefore, we need to calculate \(DU/Dt = (U_2 - U_1)/\delta t\), where \(\delta t = 0.1 s\). So, \(DU/Dt = (5929.6 - 6400)/0.1 = -4704 Btu/s\). Note that the negative sign indicates the decrease in energy of the system.

04

Use the Eulerian control volume approach to evaluate DU/Dt

The Eulerian control volume approach calculates the rate of change of energy in the system by considering the material time derivative and flux terms. In this case \(DU/Dt = \partial U/\partial t + \dot{m} \cdot (h_{out} - \tilde{u})\), where \(\dot{m} = m/\delta t = 2.0 / 0.1 = 20 slugs/s\), and \(h_{out}\) is the specific enthalpy of the exiting mass which is approximately \(c_v T' = 32.0 \cdot 103 = 3296 Btu/slug\). Substituting these values gives \(DU/Dt = -4704 + 20 \cdot (3296 - 3296) = -4704 Btu/s\).

05

Compare the results

Both approaches give the same result for the rate of change of energy in the system, \(DU/Dt = -4704 Btu/s\), indicating that the system is losing energy at this rate.

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