# Work and Heat

This is a topic from Thermodynamics:

## Contents |

## Introduction

Heat is a process, not a physical substance or a property. Heat refers to the process whereby energy is transferred down a temperature gradient.

**Sign convention:**

- Heat in is positive
- Heat out is negative

Work is the energy transferred when motion is applied against an opposing or resistive force.

**Sign convention:**

- The opposite of heat
- Work in is negative
- Work out is positive

## Mathematical Definition for Work

Recall from the first year physics courses:

- W = Fx

For an infinitesimal amount of work:

- dW = Fdx
- = PA dx
- = P dV

- dW = Fdx

Therefore, for a process moving from state 1 to state 2:

Consider the following derivations using this defintion

### Work for an Isobaric Process

- Consider an isobaric process such that pressure is constant: P = c

- Performing the integral:

- The work done is the area under the graph of the PV diagram:

- Consider an isobaric process such that pressure is constant: P = c

### Work for an Polytropic Process

- Consider an polytropic process such that PV
^{n}= c

- Performing the integral:

- Consider an polytropic process such that PV

### Work for an Isothermal Process

- Consider an isothermal process such that PV = c

- Performing the integral:

- The work done is the area under the graph of the PV diagram:

- Consider an isothermal process such that PV = c

## Types of Work

When a system is open it can be susceptible to more than one form of work.

Consider the following diagram:

- The work done on/by the piston at the top of the diagram is the type of work with which we are mostly concerned with closed systems. It is known as
**Boundary Work**and is denoted W_{b}. Boundary work, in which the system boundary undergoes a net motion, is compatible with both closed systems and open systems. - The work done by the battery on the left of the diagram is called
**Electrical Work**and is denoted W_{e}. It refers to the work done in the form of electrical power on the heating element embedded in the system. - The work done by the rotor on the right of the diagram is called
**Shaft Work**and is denoted W_{s}. - The work done by the flow of mass in and out of the system seen in the bottom left and top right of the diagram is known as
**Flow Work**and is denoted W_{flow}. Flow work is the work done when a volume of fluid flows against an opposing pressure. The points of exit and entry into the system is known as ports. This type of work is not possible in a closed system because it involved a transfer of mass across the system boundary.

The boundary work, electrical work and shaft work are generally grouped as W_{sys} such that:

**W = W**becomes_{b}+ W_{e}+ W_{s}+ W_{flow}**W = W**_{sys}+ W_{flow}

### Pump Work

*Note: This type of work requires an understanding of concepts covered later in the notes and is not needed until the second law of thermodynamics has been covered.*

- For an open system there can also be pump work, which, upon application of the first law for an open system is given as follows:

- w = (P
_{in}v_{in}+ V_{in}^{2}/2 + gz_{in}) - (P_{out}v_{out}+ V_{out}^{2}/2 + gz_{out}) - (u_{out}- u_{in}- q)

- w = (P

- However it can also be shown that q
_{rev}= u_{out}- u_{in}for a reversible process, or u_{out}- u_{in}> q for an irreversible process. - Therefore pump work can be rewritten as:

- w
_{rev}= (P_{in}v_{in}+ V_{in}^{2}/2 + gz_{in}) - (P_{out}v_{out}+ V_{out}^{2}/2 + gz_{out})

- w

- Ignoring kinetic and potential energies:

- w
_{rev}= P_{in}v_{in}- P_{out}v_{out}

- w

- The flow is incompressible, and so v
_{in}= v_{out}= v - Therefore we have:

**w**_{rev}= v(P_{in}- P_{out})

If work is equal to zero, we get Bernoulli's equation:

**P**_{in}/ρ + V_{in}^{2}/2 + gz_{in}= P_{out}/ρ + V_{out}^{2}/2 + gz_{out}