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 

Flow Systems

 For flow systems, molar flow rates are used instead of number of moles are shown in the following table:

Batch Systems

Let us define the following variables:

Thus, we can use the above definitions to describe the total moles in a batch system:

Let us assume a reaction between reactants A and B to produce products C and D occurring in a constant volume batch reactor. With A as the limiting reactant and the basis of the reaction:

A table used to compute the changes and remaining quantities of each substance in the reaction in a constant volume batch reactor can be developed as follows:

Using the mole fraction definition of Θ, for a constant volume batch reactor:

This formula is for an arbitrary species i ≠ A, Species A is the limiting reagent and i in the numerator represents the stoichiometric coefficient of species i. For the ± , the addition represents generation of products and the subtraction is for the consumption of reactants.

The stoichiometric coefficient is defined as follows where for a substance I with stoichiometric number i. It is positive for products and negative for reactants:

In a gas-phase reaction, the constant-volume condition tends to exist when n moles of reactant form n moles of product and when there is no change in temperature or pressure (i.e., ideal gas law states that volume is unchanged).

In liquid-phase reaction, the solvent dominates the solution, hence the density of the solute insignificantly impacts the system hence making most liquid-phase reactions essentially constant-volume.

Short Notes #3

 The following is a summary of reactor design equations discussed in previous entries:

PVT DIAGRAMS FOR PURE SUBSTANCES

 The Temperature-Specific Volume Diagram

To obtain this diagram we repeat the process discussed in the last blog entry (Click Here) at different pressure values. The resulting curves for water look as shown below:

It can be observed that with increasing pressure the horizontal line connecting saturated liquid and saturated vapor states becomes shorter. The reason is that as pressure increases, the specific volume of saturated liquid increases and the specific volume of the saturated vapor decreases. At P = 22.09MPa, the horizontal line between the saturated liquid and vapor states shrinks to a point at which the constant pressure line forms an inflection point with a slope = 0. This point is referred to as the critical point.

At a critical point, the saturated liquid and saturated vapor states are identical and the temperature, pressure and specific volume of a substance at this point are called the critical temperature, critical pressure and critical volume, respectively. When the pressure is above the critical pressure, a liquid and vapor phase of a pure substance does not exist in equilibrium.

The saturated liquid states can be connected by a line called the saturated liquid line while saturated vapor states are connected by a line called the saturated vapor line. These two lines meet each other at the critical point, forming a dome as shown:

All the subcooled liquid states are located in the region to the left of the saturated liquid line ad is referred to as the subcooled liquid region.

All the superheated vapor states are located to the right of the saturated vapor line and this is called the superheated vapor region.

In these two regions locates outside the dome, a pure substance exists either in liquid or vapor phase (single phase)

The region under the dome is called the saturated liquid-vapor mixture region where the liquid and vapor phases are in equilibrium.

The Pressure-Specific Volume Diagram

For a pure substance, the Pressure-Specific Volume diagram is similar to that of Temperature-Specific Volume diagram, however, the isotherms (constant temperature) lines have a downward trend as can be seen below:

The Pressure-Temperature (P-T) Diagram

Pure substances can exist as solids, liquids or as a vapor. The P-T diagram is a graphical method of showing the effects of pressure and temperature on the phases of a pure substance. It is referred to as the phase diagram with the three phases separated from one another by three lines as shown:

The curve that separates the solid and vapor phases is called the sublimation curve, and along it the solid and vapor phases are in equilibrium. The slope of the sublimation curve gives the rate of change of sublimation pressure of the solid with temperature.

The curve that separates the solid and liquid phases is called the fusion (or melting) curve, and along it the solid and liquid phases are in equilibrium. Its slope gives the rate of change of melting or freezing of solid with temperature. The fusion curve has a positive slope for most substances but water has a negative slope.

The curve that separates the liquid and vapor phases is called the vaporization curve, and along it the vapor and liquid phases are in equilibrium. Its slope gives the rate of change of vapor pressure of liquid with temperature. This curve ends at the critical temperature and pressure of the substance.

At temperatures and pressures higher than the critical values, substances are called supercritical fluids i.e., they exist in the fluid (or supercritical) region. They possess both the gaseous properties (viscosity, diffusivity, surface tension) of being able to easily diffuse into substances, and the liquid property (density) of being able to dissolve substances.

When P < P_c, a substance in the gaseous state is called either a gas (T > T_c) or a vapor (T < T_c). Under isothermal conditions, while a vapor can be liquefied by exerting pressure, a gas cannot be liquefied regardless of what pressure is applied to it. That is, a pure gas cannot be liquefied at temperatures above its critical temperature no matter what pressure is applied to it.

On the phase diagram, the point where the solid, liquid, and vapor phases coexist in equilibrium is called the triple point. This is where the liquid-vapor (vapor pressure curve), solid-liquid (fusion or melting curve), and solid-vapor (sublimation pressure curve) coexistence curves intersect. The number of degrees of freedom at the triple point is zero.

Since the fusion curve generally has a very steep slope, the triple point temperature for most substances is close to their melting (or freezing) temperature at atmospheric pressure and this is known as the normal melting (or freezing) point.

 

Phase Change of a Pure Substance

A pure substance is a substance that has a fixed chemical composition throughout. It can exist in more than one phase, but the chemical composition must be the same in all phases. An example of this is ice-liquid water mixture and liquid water-steam mixture.

Scenario: Let us consider a system consisting 1 Kg of liquid water in a piston cylinder apparatus as shown below. Assume that the ambient pressure and the piston weight maintain the cylinders pressure at 0.125MPa with the initial temperature at 25°C.

At the initial conditions (P = 0.125MPa, T = 25°C), water is a called subcooled liquid i.e., it will not vaporize if heat is transferred to the system. 

When heat is transferred to the water, its temperature increases significantly and the specific volume increases slightly while pressure remains constant. 

When T = 105.99°C, the additional heat transfer results in the water boiling and a phase change occurs. A liquid that is about to boil is referred to as a saturated liquid. 

When the liquid is vaporizing, its pressure and temperature remain constant but its specific volume increases. At this point it is called a saturated liquid-vapour mixture where both liquid and vapour phases coexist in equilibrium. 

When all liquid vapourizes, only vapour exists in the cylinder and is called saturated vapour. Any heat loss from a saturated vapour leads to condensation. 

An increase in heat into a saturated vapour leads to an increase in both temperature and specific volume and its called a superheated vapour.

  

This whole water heating process at a constant pressure can be represented on a T - Ṽ diagram as shown below:

• State 1 is the initial subcooled liquid state

• State 2 is the saturated liquid state. A saturated liquid is ready to boil with the addition of heat, and T = 105.99°C represents the boiling point temperature.

• State 4 represents a saturated vapor. A saturated vapor is ready to condense with the removal of heat, and T = 105.99°C represents the dew point temperature.

• The horizontal line joining states 2 and 4 represents an isobaric and isothermal process where phase change from liquid to vapor, or vice versa, occurs. During phase change the liquid and vapor phases are in equilibrium with each other and, as a result, both temperature and pressure remain constant.

• The line joining the states 4 and 5 represents the process in which the steam is superheated at constant pressure.

At any given pressure, the temperature at which a pure substance boils is referred to as the saturation temperature (T^sat). While at any given temperature, the pressure at which a pure substance boils is referred to as the vapour or saturation pressure (P^vap). For a pure substance, there is a definite relation between the vapour pressure and the saturation temperature which results in a vapour pressure curve shown below. The boiling of a pure component starts when the P^vap = Ambient Pressure. This explains why water boils at a temperature less than 100°C at a mountain top.