PROPERTY TABLES

For most substances, the thermodynamic properties are related to one another in a complex manner and cannot be expressed by simple equations. Thus, the properties are presented in tabulated form.

Some thermodynamic properties are easy to measure while others aren’t measured directly and need to be calculated using the relations between them and the measurable properties. The results of these measurements and calculations are then presented in tables. Some of the major properties are discussed below:

Enthalpy

In an open system, Enthalpy (H) can be defined as the amount of energy transferred across a system boundary by a moving flow. It is expressed as follows;

H = U + PV

The term PV represents the flow work and has the units of energy. Thus, H also has the units of energy. Enthalpy is a state function since U, P and V are all state functions, any combination of them must also be a state function. Enthalpy is also an extensive property. Other expressions of enthalpy are as follows;

Saturated Liquid and Saturated Vapor

The properties of saturated liquid and saturated vapor are usually presented in tabulated form and presented in appendices of many Thermodynamics books. These tables are usually either listed as either temperature and pressure tables. Thus, it is convenient to use Saturated Water: Temperature Table when temperature is given, and Saturated Water: Pressure Table when pressure is given.

The superscript L is used to denote the properties of a saturated liquid, and the superscript V to denote the properties of saturated vapor. The difference between the saturated vapor and saturated liquid states are designated by Δ, i.e.,

or;

The quality of a saturated liquid is 0, the quality of a saturated vapor is 1.

It is important to note that quality has no meaning in the subcooled (or, compressed) liquid and superheated vapor regions.

In the two-phase region, application of Gibbs phase rule gives, 2 + F = 1 + 2, thus F = 1. Hence, if one of the independent intensive variables of a two-phase mixture is known it is possible to specify the state of the system.

Superheated Vapor

In the region to the right of the saturated vapor line, a pure substance exists as superheated vapor. In this region

P < P^vap at a given temperature

T > T^sat at a given pressure

This is represented graphically as shown below:

The degrees of superheat is defined as the temperature in excess of the saturation temperature at a given pressure, i.e.,

Degrees of Superheat = T - T^sat

For a superheated vapor, application of Gibbs phase rule gives 1 + F = 1+2 ⇒ F = 2

Thus, two independent intensive variables are required to specify the state of a superheated vapor.

Subcooled (Compressed) Liquid

In the region to the left of the saturated liquid line, a pure substance exists as subcooled (or, compressed) liquid. In this region

P > P^vap at a given temperature (compressed liquid)

T < T^sat at a given pressure (subcooled liquid)

This is represented graphically as shown below:

Note: Compressed liquid implies that the pressure is greater than the saturation pressure for a given temperature. Conversely, subcooled liquid implies that the temperature is lower than the saturation temperature for the given pressure.

As an approximation, properties of compressed liquid are equal to that of a saturated liquid at the given temperature. For a compressed liquid, application of Gibbs phase rule gives 1 + F = 1+2 ⇒ F = 2

Thus, two independent intensive variables are required to specify the state of a compressed (or subcooled) liquid.

P – T Diagram Exercises

 Sketch the following processes on the P -T diagram from an original state (1) to the resulting state (2):

a) The sublimation of dry ice (solid carbon dioxide),

b) A constant pressure cylinder of superheated vapor is cooled until liquid just begins to form,

c) A constant pressure cylinder of superheated vapor is cooled until all the vapor is gone,

d) A liquid-vapor two-phase mixture is heated at constant volume until its quality is 1.0

Elementary Gas Phase Reaction in a PFR and Batch Reactor

 

Elementary Gas Phase Reaction in a CSTR

 

Introduction to Heat Transfer

Heat Transfer (HT) is the thermal energy in transit due to a temperature difference. This means that the existence of temperature difference leads to HT occurring in a medium or between media.

There are different types or modes of HT processes as listed below:

• Conduction – HT that occurs due to the existence of a temperature gradient in a stationary medium, either solid or fluid.
• Convection – HT that occurs between a surface and a moving fluid at different temperatures.
• Thermal Radiation – HT that occurs between two surfaces at different temperatures in the absence of an intervening medium. This is due to the fact that all surfaces of finite temperature emit energy in the form of electromagnetic waves. 

The HT processes can be quantified using rate equations to compute the amount of energy transferred per unit time.

a) Conduction

Can be viewed as the transfer of energy from a more energetic particles in a substance to less energetic ones due to interactions between the particles. Conduction is therefore heat transfer (HT) through a substance that has no bulk (macroscopic) motion e.g., solids.

A high temperature means high molecular energy thus neighboring molecules collide; energy is transferred to the less energetic molecules. Thus, when a temperature gradient exists, energy transfer by conduction occurs in the direction of decreasing temperature.

For heat conduction, the rate equation is known as Fourier’s Law of Heat Conduction.

For a 1 – dimensional plane wall shown above, having a temperature distribution T(x), the rate equation is expressed as follows:

Where:

q_x = heat flux in x-direction

k = thermal conductivity (W/m.K)  [this is a characteristic of the wall material]

dT/dx = temperature gradient

If k is a constant, the heat flux can be calculated as:

The negative sign is due to HT occurring in the direction of decreasing temperature.

Under steady state conditions, the temperature distribution is linear and can be expressed as:

The heat flux can therefore be simplified into:

Or

NOTE: The above equation is the rate of HT per unit area, thus the heat rate by conduction through a plane wall of Area, A, is the product of the heat flux and the area. 

b) Convection

This is heat transfer (HT) between a surface and a moving fluid at different temperatures.

It is comprised of two mechanisms:

• Energy transfer due to random molecular motion (diffusion).
• Energy transfer by the bulk or macroscopic, motion of the fluid. This motion in the presence of a temperature gradient contributes to HT.

Convection is used when referring to the cumulative transport, while advection is the transport due to bulk fluid motion.

A region in the fluid develops where the fluid velocity varies from zero at the surface to a finite value u_∞ at the outer region of the fluid. This region is known as a boundary or hydrodynamic layer.

Also, if the temperature of the surface varies from the flow temperature, the fluid region will have varying temperature from T_s at the surface (y=0) to T_∞ at the outer region of the fluid. This is the thermal boundary layer. If T_s > T_∞ convective HT occurs from the surface to outer flow.

At the surface (y=0), fluid velocity is zero and heat us transferred only by random molecular motion (diffusion) while bulk fluid motion increases as the boundary layer grows.

Convection can be classified according to nature of flow:

• Forced convection - When flow is caused by external means like fans, pumps, wind.
• Free (natural) convection - Flow occurs due to buoyancy forces which are due to density differences caused by the variation in fluid temperature.
• Mixed forced and natural convection.
• Boiling and condensation - For typical convection, energy transferred is the sensible or internal thermal energy, but for these two, there's an addition of latent heat exchange due to phase change.

Below is a table summarizing the heat transfer coefficient of convection processes: 

The general rate equation for convection is known as Newton’s law of cooling and is represented as:

Where:

q_x = convective heat flux (W/m²)

T_s = Surface temperature (K)

T_∞ = Fluid temperature (K)

h = Convection HT coefficient (W/m².K)

c) Radiation

Thermal radiation is the energy emitted by matter at a non-zero temperature. The energy of the radiation is transferred by electromagnetic waves.

While conduction and convection need material medium for heat transfer (HT), radiation can occur efficiently in a vacuum. 

Radiation emitted by the surface originated from the thermal energy of matter bound by the surface as shown below:

The rate at which energy is released per unit area (W/m²) is termed as the surface emittive power, E. For an ideal emitter, this is expressed by the Steffan-Boltzmann law and such a surface is called an ideal radiator or blackbody.

Where:

T = Absolute temperature of the surface (K)

σ = Stefan-Boltzmann constant (σ = 5.67 x 10^-8 W/m⁴.K⁴)

However, for a real surface heat flux emitted is less than that of a blackbody. A radioactive property of the surface known as emissivity, ε. Is used to modify the above equation into the following:

Emissivity is in the range 0≤ ε ≤1 and provides a measure of how efficiently a surface emits energy relative to a blackbody.

Irradiation (G) is the radiation from any source like the sun, that is incident on a unit area of the surface. This radiation can be absorbed, reflected or transmitted by the surface.

Absorptivity (α), is the rate at which radiant energy is absorbed per unit surface area and is in the range 0≤ α ≤1. It can be expressed as:

If α < 1 and surface is opaque, then portions of irradiation are reflected. If surface is semitransparent, portions of irradiation is transmitted.

A special case that occurs frequently involves radiation exchange between a small surface at T_s and a much larger, isothermal surface that completely surrounds the smaller one a T_surr.

When α = ε the surface is called a gray surface.

Irradiation can be approximated by emission from a blackbody at T_surr in which case, G = σT_surr⁴.

For a gray surface, the net rate of radiation HT from the surface per unit area is:

The above equation represents the difference between thermal energy released due to radiation emission and that gained due to radiation absorption. 

 

Short Notes #5

Avogadro’s Number

The Avogadro Number is the proportionality factor that relates the number of constituent particles (molecules, atoms or ions) contained in one mole of a substance. Its SI unit is the reciprocal mole, and is exactly 6.02×10²³ mol⁻¹.

Robert Millikan, an American Physicist, was the first to measure the charge on an electron which helped determine the Avogadro’s Number. The electron charge is measured as 1.6021765 x 10^-19 coulombs per electron. The Faraday is the charge on a mole of electrons and is estimated as 96,485.34 coulombs per mole of electron. The Avogadro’s Number is then obtained by dividing the charge on a mole of electrons by the charge on a single electron i.e.,

Avogadro’s Number = (charge on a mole of electrons / charge on a single electron)

                                  = (96,485.34) / 1.6021765 x 10^-19)

                                  = 6.02 x 10²³ particles per mole

Short Notes #4

The following summarizations can be helpful when designing isothermal reactors.

The concentration of a species in a chemical reaction can be expressed as a function of conversion using the following algorithm. 

The following is an algorithm that can be followed when designing an Isothermal-reaction for conversion.

Algorithm for isothermal reactors is summarized below. 

(Source: Elements of Chemical Reaction Engineering (5th Edition) by H. Scott Fogler)