Question

A closed system undergoes a process during which there is energy transfer from the system by heat at a constant rate of \(10 \mathrm{~kW}\), and the power varies with time according to $$ \dot{W}= \begin{cases}-8 t & 01 \mathrm{~h}\end{cases} $$ where \(t\) is time, in \(\mathrm{h}\), and \(\dot{W}\) is in \(\mathrm{kW}\). (a) What is the time rate of change of system energy at \(t=\) \(0.6 \mathrm{~h}\), in \(\mathrm{kW}\) ? (b) Determine the change in system energy after \(2 \mathrm{~h}\), in \(\mathrm{kJ}\).

Short answer

a) 14.8 kW \ b) 115200 kJ

Step-by-step solution

Understand the given data

The system undergoes a process with energy transfer through heat at a constant rate of 10 kW. The power \( \dot{W} \) varies with time as follows: \[ \dot{W} = \begin{cases}\-8t & 0 < t \leq 1 \mathrm{h} \ \ \ \ \ \ -8 & t > 1 \ \mathrm{h}\end{cases} \] Given \( \dot{W} \) is in kW and \( t \) is in hours.

Calculate the time rate of change of system energy at t = 0.6 h

The rate of energy transfer by heat is \(Q = 10 \mathrm{kW} \). At \( t = 0.6 \mathrm{h} \), the power \( \dot{W} = -8t \). For \( t = 0.6 \mathrm{h} \), \ \( \dot{W} = -8 \cdot 0.6 = -4.8 \mathrm{kW} \). The time rate of change of system energy is given by \( \frac{dE_{\rm system}}{dt} = \dot{Q} - \dot{W} \). Substituting the values we get: \ \( \frac{dE_{\rm system}}{dt} = 10 - (-4.8) = 10 + 4.8 = 14.8 \mathrm{kW} \).

Determine the change in system energy after 2 hours

The total change in system energy can be determined by integrating the rate of change over the time periods. For 0 < t \leq 1 \mathrm{h}, \ \(\ \frac{dE_{\rm system}}{dt} = 10 - (-8t) = 10 + 8t\ . \) Integrate \(10 + 8t \) from 0 to 1 hour: \[ \Delta E_{\rm system,1} = \int_0^1 (10 + 8t) \ dt = \[ 10t + 4t^2 \]_{0}^{1} = 10 \cdot 1 + 4 \cdot 1^2 = 10 + 4 = 14 \ \mathrm{kWh} \]. For t > 1 hour: \ \(\ \frac{dE_{\rm system}}{dt} = 10 - (-8) = 10 + 8 = 18 \ \mathrm{kW} \) \ (constant). Integrate 18 from 1 to 2 hours: \[ \Delta E_{\rm system,2} = \int_1^2 18 \ dt = 18t \[ 1 \to 2 \] = 18 \cdot 1 = 18 \ \mathrm{kWh} \]. Total change in system energy after 2 hours: \ \(\ \Delta E_{\rm system,tot} = 14 \ \mathrm{kWh} + 18 \ \mathrm{kWh} = 32 \[ \mathrm{kWh} \] = 32 \cdot 3600\ \ve 115200 \mathrm{kJ} \).

Key concepts

closed system

In thermodynamics, systems can be classified based on their ability to exchange energy and matter with their surroundings. A closed system cannot exchange matter with its surroundings but can exchange energy in the form of heat or work. This means the mass remains constant, but energy transfer can still occur, impacting the system's internal energy. For the given exercise, the focus is on energy transfer in and out of the system through heat and work, impacting the system's internal energy.

rate of energy transfer

The rate of energy transfer is essential in understanding how energy changes in a system over time. In our exercise, heat is transferred out of the system at a constant rate of 10 kW. This steady rate signifies a continuous energy loss due to heat. On the other hand, the power function changes with time, influencing the system's energy at different moments. This timed variation in power affects the rate at which work is done on or by the system, making it a critical aspect when calculating energy changes. For instance, at t = 0.6 h, power equals -4.8 kW, indicating the energy transfer rate affecting the closed system’s total energy.

energy integration

Energy integration involves calculating the total energy change over time by integrating the rate of energy transfer. When dealing with time-varying power, like in our exercise, each segment of time needs to be accounted for uniquely. For the interval 0 < t ≤ 1 h, the rate of change of system energy is integrated as 10 + 8t. The integration over this interval yields a total change in energy for that period. Similarly, for t > 1 h, the constant energy rate of 18 kW simplifies the integration process across that period. Summing these integrated values gives the total energy change after 2 hours, revealing how energy accumulates or depletes in the system over time.

time-dependent power function

A time-dependent power function means that the power input or output changes based on time. In our example, the function is defined as -8t for 0 < t ≤ 1 h and -8 for t > 1 h. This indicates that power output initially ramps up with time until it reaches a steady value. This changing power impacts the energy calculations over different intervals. At early times, the power varies linearly with time, requiring careful integration to capture the energy change accurately. After 1 hour, the power remains constant, simplifying further calculations but requiring an understanding of its implications on the system’s energy over time.