Question

As shown in Fig. P2.37, \(5 \mathrm{~kg}\) of steam contained within a piston- cylinder assembly undergoes an expansion from state 1 , where the specific internal energy is \(u_{1}=2709.9 \mathrm{~kJ} / \mathrm{kg}\), to state 2 , where \(u_{2}=2659.6 \mathrm{~kJ} / \mathrm{kg}\). During the process, there is heat transfer to the steam with a magnitude of \(80 \mathrm{~kJ}\). Also, a paddle wheel transfers energy to the steam by work in the amount of \(18.5 \mathrm{~kJ}\). There is no significant change in the kinetic or potential energy of the steam. Determine the energy transfer by work from the steam to the piston during - the Qrocet in kJ.

Short answer

The work done by the steam on the piston is 313 kJ.

Step-by-step solution

Understand the given parameters

First, identify and note the given parameters:- Mass of steam: 5 kg- Specific internal energy at state 1: \(u_{1}=2709.9 \text{ kJ/kg} \)- Specific internal energy at state 2: \(u_{2}=2659.6 \text{ kJ/kg} \)- Heat transfer to the steam: \(Q = 80 \text{ kJ} \)- Work done by the paddle wheel: \(W_{paddle} = 18.5 \text{ kJ} \)

Apply the First Law of Thermodynamics

The First Law of Thermodynamics is expressed as:\[ \text{Change in internal energy} = \text{Heat added to the system} - \text{Work done by the system} \]Or in equation form:\[ \text{Δ}U = Q - W \]

Calculate the change in internal energy

The change in internal energy (\( \text{Δ}U \)): \[ \text{Δ}U = m(u_2 - u_1) \]Given, mass \( m = 5 \text{ kg} \)\[ u_1 = 2709.9 \text{ kJ/kg}, \, u_2 = 2659.6 \text{ kJ/kg} \]Substitute these values:\[ \text{Δ}U = 5 \times (2659.6 - 2709.9) = 5 \times (-50.3) = -251.5 \text{ kJ} \]

Rearrange the First Law equation

Rearrange the First Law of Thermodynamics equation to solve for the work done by the steam on the piston (\( W_{piston} \)):\[ \text{Δ}U = Q - W_{\text{total}} \]Where:\[ W_{\text{total}} = W_{\text{paddle}} + W_{\text{piston}} \]We get:\[ -251.5 = 80 - (18.5 + W_{\text{piston}}) \]

Solve for the work done by the steam on the piston

Now solve for \( W_{\text{piston}} \):\[ -251.5 = 80 - 18.5 - W_{\text{piston}} \]\[ -251.5 = 61.5 - W_{\text{piston}} \]\[ W_{\text{piston}} = 61.5 + 251.5 = 313 \text{ kJ} \]

Key concepts

Internal Energy

Internal energy is the total energy contained within a system. It encompasses the kinetic and potential energies of all the particles in that system. In thermodynamics, internal energy is usually denoted as U.

Heat Transfer

Heat transfer is the process of energy moving from one body or substance to another due to a temperature difference. It can occur in three modes: conduction, convection, and radiation.

Work Done

In thermodynamics, work done is a measure of energy transfer that occurs when an external force moves an object. It is often represented by the symbol W. One common example of work done is in piston-cylinder assemblies where the expansion or compression of gas performs work on the piston.