Contents

## Heat and Work

We’ve seen that the internal energy changes with **Q**, which is the **net heat added** to the system and **W**, which is the **net work done by** the system. We now examine how the work done by and the heat added to the system during a thermodynamic process depend on the details of how the process takes place.

## Heat in Thermodynamics

While **internal energy** refers to the total energy of all the molecules within the object, **heat** is the amount of energy** flowing** from one body to another spontaneously due to their temperature difference. **Heat** is a form of energy, but it is **energy in transit**. Heat is not a property of a system. However, the transfer of energy as heat occurs at the molecular level as a result of a **temperature difference**.

Consider a **block of metal** at high temperature, that consists of atoms that are oscillating intensely around their average positions. **At low temperatures**, the atoms continue to oscillate, but with **less intensity**. If a hotter block of metal is put in contact with a cooler block, the intensely oscillating atoms at the edge of the hotter block gives off its kinetic energy to the less oscillating atoms at the edge of the cool block. In this case there is **energy transfer** between these two blocks and **heat flows** from the hotter to the cooler block by this random vibrations.

In general, when two objects are brought into** thermal contact**, **heat will flow** between them **until** they come into **equilibrium** with each other. When a **temperature difference** does exist heat flows spontaneously **from the warmer system to the colder system**. Heat transfer occurs by **conduction** or by **thermal radiation**. When the **flow of heat stops**, they are said to be at the** same temperature**. They are then said to be in **thermal equilibrium**.

As with work, the amount of heat transferred **depends upon the path** and not simply on the initial and final conditions of the system. There are actually many ways to take the gas from state i to state f.

Also, as with work, it is important to distinguish between **heat added** to a system from its surroundings and **heat removed** from a system to its surroundings. Q is positive for heat added to the system, so if heat leaves the system, Q is negative. Because W in the equation is the work done by the system, then if work is done on the system, W will be negative and E_{int} will increase.

The symbol **q** is sometimes used to indicate the heat added to or removed from a system **per unit mass**. It equals the total heat (Q) added or removed divided by the mass (m).

## Heat Capacity

**Different substances** are affected to **different magnitudes** by the **addition of heat**. When a given amount of heat is added to different substances, their temperatures increase by different amounts. This **proportionality constant** between the **heat Q** that the object absorbs or loses and the resulting **temperature change T** of the object is known as the **heat capacity C** of an object.

**C = Q / ΔT**

**Heat capacity** is an extensive property of matter, meaning it is proportional to the size of the system. **Heat capacity C** has the unit of energy per degree or energy per kelvin. When expressing the same phenomenon as an intensive property, the **heat capacity** is divided by the amount of substance, mass, or volume, thus the quantity is independent of the size or extent of the sample.

## Specific Heat Capacity

The **heat capacity** of a substance per unit mass is called the **specific heat capacity (c _{p})** of the substance. The subscript p indicates that the heat capacity and

**specific heat capacity**apply when the heat is added or removed

**at constant pressure**.

*c _{p} = Q / mΔT*

## Specific Heat Capacity

In the Ideal Gas Model, the **intensive properties c_{v}** and

*are defined for pure, simple compressible substances as partial derivatives of the*

**c**_{p}**internal energy**and

*u(T, v)***enthalpy**, respectively:

*h(T, p)*where the subscripts **v** and **p** denote the variables held fixed during differentiation. The properties **c _{v} **and

**c**are referred to as

_{p}**specific heats**(or

**heat capacities**) because under certain special conditions they relate the temperature change of a system to the amount of energy added by heat transfer. Their SI units are

**J/kg K**or

**J/mol K**. Two specific heats are defined for gases, one for

**constant volume (c**and one for

_{v})**constant pressure (c**.

_{p})According to the **first law of thermodynamics**, for constant volume process with a monatomic ideal gas the molar specific heat will be:

*C _{v} = 3/2R = 12.5 J/mol K*

because

*U = 3/2nRT*

It can be derived that the **molar specific heat** at constant pressure is:

**C _{p} = C_{v} + R = 5/2R = 20.8 J/mol K**

This ** C_{p}** is greater than the molar specific heat at constant volume

**, because energy must now be supplied**

*C*_{v}**not only**to

**raise the temperature**of the gas but also for the

**gas to do work**because in this case volume changes.

## Latent Heat of Vaporization

In general, when a material **changes phase** from solid to liquid, or from liquid to gas a certain amount of energy is involved in this change of phase. In case of liquid to gas phase change, this amount of energy is known as the **enthalpy of vaporization**, (symbol ∆H_{vap}; unit: J) also known as the **(latent) heat of vaporization** or heat of evaporation. Latent heat is the amount of heat added to or removed from a substance to produce a change in phase. This energy breaks down the intermolecular attractive forces, and also must provide the energy necessary to expand the gas (the **pΔV work**). When latent heat is added, no temperature change occurs. The enthalpy of vaporization is a function of the pressure at which that transformation takes place.

Latent heat of vaporization – water at 0.1 MPa (atmospheric pressure)

**h _{lg} = 2257 kJ/kg**

Latent heat of vaporization – water at 3 MPa (pressure inside a steam generator)

**h _{lg} = 1795 kJ/kg**

Latent heat of vaporization – water at 16 MPa (pressure inside a pressurizer)

**h _{lg} = 931 kJ/kg**

The **heat of vaporization** diminishes with increasing pressure, while the boiling point increases. It vanishes completely at a certain point called the critical point. Above the critical point, the liquid and vapor phases are indistinguishable, and the substance is called a supercritical fluid.

The heat of vaporization is the heat required to completely vaporize a unit of saturated liquid (or condense a unit mass of saturated vapor) and it equal to **h _{lg} = h_{g} − h_{l}**.

The heat that is necessary to melt (or freeze) a unit mass at the substance at constant pressure is the heat of fusion and is equal to **h _{sl} = h_{l} − h_{s}**, where h

_{s }is the enthalpy of saturated solid and h

_{l}is the enthalpy of saturated liquid.

## Work in Thermodynamics

In thermodynamics, **work** performed by a system is the energy transferred by the system to its surroundings. Kinetic energy, potential energy and internal energy are forms of energy that are properties of a system. **Work is a form of energy**, but it is **energy in transit**. A system contains no work, work is a process done by or on a system. In general, work is defined for mechanical systems as the action of a force on an object through a distance.

*W = F . d*

where:

W = work (J)

F = force (N)

d = displacement (m)

## pΔV Work

Pressure-volume work (or *pΔV ***Work**) occurs when the volume *V* of a system changes. The *pΔV ***Work** is equal to the area under the process curve plotted on the pressure-volume diagram. It is known also as the **boundary work**. ** ****Boundary work** occurs because the mass of the substance contained within the system boundary causes a force, the pressure times the surface area, to act on the boundary surface and make it move. **Boundary work** (or **pΔV*** ***Work**) occurs when the **volume**** ****V**** ****of a system changes**. It is used for calculating piston displacement work in a **closed system**. This is what happens when **steam**, or gas contained in a piston-cylinder device expands against the piston and forces the piston to move.

**Example:**

Consider a frictionless piston that is used to provide a constant pressure of **500 kPa** in a cylinder containing steam (superheated steam) of a volume of **2 m**** ^{3 }** at

**500 K**.

Calculate the final temperature, if **3000 kJ** of** heat** is added.

**Solution:**

Using steam tables we know, that the **specific enthalpy** of such steam (500 kPa; 500 K) is about** 2912 kJ/kg**. Since at this condition the steam has density of 2.2 kg/m^{3}, then we know there is about **4.4 kg of steam** in the piston at enthalpy of 2912 kJ/kg x 4.4 kg =** 12812 kJ**.

When we use simply **Q = H**_{2}** − H**** _{1}**, then the resulting enthalpy of steam will be:

H_{2} = H_{1} + Q = **15812 kJ**

From **steam tables**, such superheated steam (15812/4.4 = 3593 kJ/kg) will have a temperature of **828 K (555°C)**. Since at this enthalpy the steam have density of 1.31 kg/m^{3}, it is obvious that it has expanded by about 2.2/1.31 = 1.67 (+67%). Therefore the resulting volume is 2 m^{3} x 1.67 = 3.34 m^{3} and ∆V = 3.34 m^{3} – 2 m^{3} = 1.34 m^{3}.

The **p∆V **part of enthalpy, i.e. the work done is:

**W = p∆V = 500 000 Pa x 1.34 m**^{3}** = 670 kJ**

———–

During the **volume change**, the **pressure** and **temperature** may also change. To calculate such processes, we would need to know how pressure varies with volume for the actual process by which the system changes **from state i to state f**. The **first law of thermodynamics** and the work can then be expressed as:

When a thermodynamic system changes from an** initial state** to a **final state**, it passes through a **series of intermediate states**. We call this series of states a **path**. There are always infinitely many different possibilities for these intermediate states. When they are all equilibrium states, the path can be plotted on a **pV-diagram**. One of the most important conclusions is that:

*The work done by the system depends not only on the initial and final states, but also on the intermediate states—that is, on the path.*

**Q and W are path dependent, whereas ΔE**_{int}** is path independent. **As can be seen from the picture (p-V diagram), work is path dependent variable. The blue area represents the ** pΔV Work** done by a system as it goes from an initial state i to a final state f. Work W is positive because the system’s volume increases. The second process shows, that work is greater and that depends on path of the process.

Moreover, we can take the system through a series of states forming a **closed loop**, such **i ⇒ f ⇒ i**. In this case the **final state is the same as the initial state**, but the **total work done** by the system **is not zero**. A positive value for work indicates that work is done by the system on its surroundings. A negative value indicates that work is done on the system by its surroundings.

## Example: Turbine Specific Work

A **high-pressure stage** of **steam turbine** operates at steady state with inlet conditions of **6 MPa**, **t = 275.6°C**, x = 1 (point C). Steam leaves this stage of turbine at a pressure of **1.15 MPa**, **186°C** and **x = 0.87** (point D). Calculate the enthalpy difference between these two states. Determine the specific work transfer.

The enthalpy for the state C can be picked directly from** steam tables**, whereas the enthalpy for the state D must be calculated using vapor quality:

*h*_{1, wet}* = ***2785 kJ/kg**

*h*_{2, wet}* = h*_{2,s}* x + (1 – x ) h***_{2,l}** = 2782 . 0.87 + (1 – 0.87) . 790 = 2420 + 103 =

**2523 kJ/kg**

**Δh = 262 kJ/kg**

Since in adiabatic process **dh = dw**, **Δh = 262 kJ/kg is **the **turbine specific work**.