FICKS LAW OF DIFFUSION

Fick's law of diffusion

• Mass transfer is the natural tendency to transfer a given component (species) in a mixture from a region of high concentration to a region of low concentration to bring about a uniform or equilibrium condition. 
• The mass transfer has three requirements: 

1. that transfer occurs only in a mixture,
2. that at least one substance within the mixture moves from region of high concentration to a region of low concentration,
3. that the rate of mass transfer—i.e., the “flux” of a given substance—be proportional to the concentration gradient of that substance. 

Fick's  law

• In 1855 FICK proposed a relation between the flux of the diffusing substance and concentration gradient as the first law of diffusion which named as Fick's law of diffusion.

Fick’s law states that the flux of a diffusing component A in the z-direction in a binary mixture of A and B is proportional to the molar concentration gradient.

• Hence Fick's law of diffusion for component A in a binary mixture of A and B for steady-state diffusion is

           Where

• The negative sign in the equation indicates that diffusion occurs in the direction of decrease in concentration. Hence the term dCA/dz is -ve and flux become positive
• JA = molar flux of A in the z-direction [kmol/(m2.s)]
• CA = molar concentration of A [ kmol/ m3]
• dCA/dz = concentration gradient in the z-direction
• DAB = proportionality constant, diffusion coefficient for component A diffusing through B [m2/s]
• Z = distance in the direction of diffusion

COMPARISION OF SINGLE AND MULTIPLE EFFECT EVAPORATOR

Comparison of single  and multiple effect evaporator

Single-effect evaporators 

• Single-effect evaporators are used 

1. when the throughput is low, 
2. when a cheap supply of steam is available, 
3. when expensive materials of construction must be used as is the case with corrosive feedstocks 
4. when the vapour is so contaminated so that it cannot be reused. 

• The feed and saturated steam with temperature at TF and TS respectively enter the heat- exchange section 
• Condensed steam leaves as condensate or drips.
• The solution in the evaporator is assumed to be completely mixed.
• Hence, the concentrated product and the solution in the evaporator have the same composition.
• Temperature T1 is the boiling point of the solution.
• The temperature of the vapor is also T1, since it is in equilibrium with the boiling solution.
• The pressure is P1, which is the vapor pressure of the solution at T1
• If the solution to be evaporated is assumed to be dilute and like water, then 1 kg of steam condensing will evaporate approximately 1 kg of vapor (if the feed entering has TF near the boiling point) 
• Single-effect evaporators are often used when the required capacity of operation is relatively small and/or the cost of steam is relatively cheap compared to the evaporator cost.
• However, for large-capacity operation, using more than one effect will markedly reduce steam costs
• The heat requirements of single-effect continuous evaporators may be obtained from mass and energy balances. 

Multiple - effect evaporator

• The single effect evaporator uses rather more than 1 kg of steam to evaporate 1 kg of water. 
• The latent heat of the vapor leaving in single effect evaporator is not used but is discarded.
• Much of this latent heat, however, can be recovered and reused by employing a multiple - effect evaporator, that is, vapor from one effect serves as the heating medium for the next.
• The economy of the system, measured by the kilograms of water vaporized per kilogram of steam condensed, increases with the number of effects.
• in multiple effect evaporator, the pressure in each effect is lower than that of the effect to which it receives steam and higher than that of the effect to which it supplies vapors
• Each effect, in itself, act as a single effect evaporator, and each has a temperature drop across its heating surface corresponding to the pressure drop in that effect.

Forward-feed multiple - effect evaporator 

• A simplified diagram of a forward-feed triple- effect evaporation system is shown in Fig. 

• If the feed to the first effect is near the boiling point at the pressure in the first effect, 1 kg of steam will evaporate almost 1 kg of water.
• The first effect operates at a temperature that is high enough that the evaporated water serves as the heating medium to the second effect.
• Here, again, almost another kg of water is evaporated, which can then be used as the heating medium to the third effect.
• As a very rough approximation, almost 3 kg of water will be evaporated for 1 kg of steam in a three-effect evaporator.
• Hence, the steam economy, which is kg vapor evaporated/kg steam used, is increased.
• This also holds approximately more than three effects.
• However, the increased steam economy of a multiple-effect evaporator is gained at the expense of the original first cost of these evaporators
• In forward-feed operation as shown in Fig. fresh feed is added to the first effect and flows to the next in the same direction as the vapor flow.
• This method of operation is used when the feed is hot or when the final concentrated product might be damaged at high temperatures.
• The boiling temperatures decrease from effect to effect. This means that if the first effect is at P1 = 1 atm abs pressure, the last effect will be under vacuum at a pressure P3.
• The concentration of the liquid increases from the first effect to the last effects
• This pattern of liquid flow is the simplest
• It requires a pump for feeding dilute solution to the first effect, since this effect is often at about atmospheric pressure, and a pump to remove thick liquor from the last effect.
• The transfer from effect to effect, however, can be done without pumps, since the flow in the direction of decreasing pressure, and control valves in the transfer line all that is required.

Backward-feed multiple - effect evaporator 

• In the backward-feed operation shown in Fig. for a triple-effect evaporator, the fresh feed enters the last and coldest effect and continues on until the concentrated product leaves the first effect. 

• This method of reverse feed is advantageous when the fresh feed is cold since a smaller amount of liquid must be heated to the higher temperatures in the second and first effects.
• However, liquid pumps must be used in each effect, since the flow is from low to high pressure.
• This reverse-feed method is also used when the concentrated product is highly viscous.
• The high temperatures in the early effects reduce the viscosity and give reasonable heat-transfer coefficients.
• Backward feed often gives a higher capacity than forward feed when the thick liquor is viscous, but it may give a lower economy than forward feed when the feed liquor is cold.

Parallel-feed multiple-effect evaporators

• Parallel-feed in multiple-effect evaporators involves the adding of fresh feed and withdrawal of the concentrated products from each effect.
• The vapor from each effect is still used to heat the next effect.
• This method of operation is mainly used when the feed is almost saturated and solid crystals are the product, as in the evaporation of brine to make salt

MEASURES OF EVAPORATOR PERFORMANCE

There are three main measures of evaporator performance:

• Capacity (kg vaporized / time)
• Economy (kg vaporized / kg steam input)
• Steam Consumption (kg / hr)

The performance of a evaporator is evaluated by the capacity and the economy.

Economy

• Economy is the number of kg of water vaporized per kg of steam fed to the unit.
• The rate of heat transfer q through the heating surface of an evaporator, by the definition of overall heat transfer coefficient, is product of three factors

1. The area of heat transfer surface A
2. The overall heat transfer coefficient U
3. The overall temperature drop ΔT

Q = U * A * ΔT

• Economy calculations are determined using enthalpy balances.
• The key factor in determining the economy of an evaporator is the number of effects.
• The economy of a single effect evaporator is always less than 1.0. 
• Multiple effect evaporators have higher economy but lower capacity than single effect.
• The thermal condition of the evaporator feed has an important impact on economy and performance. 
• If the feed is not already at its boiling point, heat effects must be considered. 
• If the feed is cold (below boiling) some of the heat going into the evaporator must be used to raise the feed to boiling before evaporation can begin; this reduces the capacity.
• If the feed is above the boiling point, some flash evaporation occurs on entry.

Capacity

• Capacity is defined as the no of kilograms of water vaporized per hour.
• If the feed to the evaporator is at the boiling temperature corresponding to the absolute pressure in the vapor space, all the heat transferred through the heating surface is available for evaporation and the capacity is proportional to q.
• If the feed is cold, the heat required to heat it to its boiling point may be quite large and the capacity for a given value of q is reduced accordingly, as heat used to heat the feed is not available for evaporation.
• if the feed is at a temperature above the boiling point in the vapor space, a portion of the feed evaporates spontaneously by adiabatic equilibration with the vapor-space pressure and the capacity is greater than that corresponding to q. This process is called flash evaporation.

Steam consumption

• Steam consumption is very important to know, and can be estimated by the ratio of capacity divided by the economy. 
• That is the steam consumption (in kg/h) is

                                          Consumption = Capacity/Economy.

Heat transfer in evaporators

• The rate equation for heat transfer takes the form:

Q = U * A * ΔT

where:

1.  Q is the heat transferred per unit time
2. U is the overall coefficient of heat transfer
3. A is the heat transfer surface
4. T is the temperature difference between the two streams. 

• In applying this equation to evaporators, there may be some difficulty in deciding the correct value for the temperature difference because of what is known as the boiling point rise (BPR) or boiling point elevation (BPE)
• If water is boiled in an evaporator under a given pressure, then the temperature of the liquor may be determined from steam tables and the temperature difference is readily calculated. 
• At the same pressure, a solution has a boiling point greater than that of water, and the difference between its boiling point and that of water is the BPR or BPE. 
• For example, at atmospheric pressure (101.3 kN/m2 ), a 25 per cent solution of sodium chloride boils at 381 K and shows a BPR of 8 deg K. If steam at 389 K were used to concentrate the salt solution, the overall temperature difference would not be (389 − 373) = 16 deg K, but (389 − 381) = 8 deg K. Such solutions usually require more heat to vaporise unit mass of water, so that the reduction in capacity of a unit may be considerable. 
• The value of the BPR cannot be calculated from physical data of the liquor, though Duhring’s rule is often used to find the change in BPR with pressure. 
• Duhring’s rule states that the boiling point of given solution is a linear function of the boiling point of pure water at the same pressure.
• Thus, if the boiling point of the solution is plotted against that of water at the same pressure, then a straight line is obtained.
• Thus, if the pressure is fixed, the boiling point of water is found from steam tables, and the boiling point of the solution from Duhring’s plot.
• Different lines are obtained for different concentrations.
• The boiling point rise is much greater with strong electrolytes, such as salt and caustic soda.

TYPES OF EVAPORATOR

Types of Evaporator

Single effect evaporator

When a single evaporator is used, the vapor from the boiling liquid is condensed and discarded. This method is called single effect evaporation, and although it is simple, it utilizes steam ineffectively.

• The solution to be concentrated flows inside the tubes. 
• The heating medium is steam condensing on metal tubes. 
• Usually boiling liquid is enter under a moderate vacuum.
• This increases the temperature difference between the steam and boiling liquid.

Multiple effect evaporator

Increasing the evaporation per kilogram of steam by using a series of evaporators between the steam supply and the condenser is called multiple effect evaporator

• If vapor from one evaporator is fed into the steam chest of the second evaporator and the vapor from second is then sent to a condenser, the operation becomes the double effect. 
• The heat in the original steam is reused in the second effect, and the evaporation is achieved by a unit mass of steam in the same manner.

Once through evaporator

• In once-through evaporation, the feed liquor passes through the tube only once, releases the vapor, and leaves the unit as thick liquor. 
• All the evaporation is accomplished in a single pass. 
• The ratio of evaporation to feed is limited in single pass units: the evaporators are well adopted to multiple effect operation, where the total amount of operation can be spread over several effects. 
• Agitated film evaporators are always operated once- through  falling film and climbing film evaporators can also be operated in this way. 
• Once through evaporators are specially useful for heat-sensitive materials. 

Circulation evaporators

• In circulation evaporators a pool of liquid is held within the equipment. 
• Incoming feed mixes with the liquid from the pool, and the mixture passes through the tubes. 
• Unevaporated liquid discharged from the tubes returns to the pool so that only part of the total evaporation occurs in one pass. 
• All forced circulation evaporators are operated in this way; climbing- film evaporators are usually circulation units.
• These are adapted to single-effect evaporation. 
• These are not suited for heat-sensitive materials.

EVAPORATION

Evaporation

• Evaporation is when a liquid becomes a gas without forming bubbles inside the liquid volume.
• During evaporation, only the molecules near the liquid surface are changing from a liquid to vapor.
• Evaporation is the process that occurs when the surface of a liquid is converted into a gas
• For example, evaporation occurs when a glass of water is left out overnight and the water level is found to drop.
• Another example is when steam flows through a tube that is submerged in a pool of liquid, minutes bubbles of vapor form at random points on the surface of the tube. The heat is passing through the tube surface where no bubbles form enters the surrounding liquid by convention. Then some of the heat in the liquid then flows toward the bubbles, causing evaporation from its inner surface into itself.
• The objective of evaporation is to concentrate a solution consisting of a non volatile solute and volatile solvent.
• Evaporation is conducted by vaporizing a portion of the solvent to produce a concentrated solution of thick liquor.
• Normally, in evaporation thick liquor is the voluble product and the vapor is condensed and discarded.
• Hence, Evaporation is a widely used method for the concentration of aqueous solutions, involves the removal of water from a solution by boiling the liquor in a suitable vessel, an evaporator, and withdrawing the vapor. 
• If the solution contains dissolved solids, the resulting strong liquor may become saturated so that crystals are deposited. 
• Evaporation is achieved by adding heat to the solution to vaporize the solvent. 
• The heat is supplied principally to provide the latent heat of vaporization, and, by adopting methods for recovery of heat from the vapor, it has been possible to achieve a great economy in heat utilization. 
• Whilst the normal heating medium is generally low-pressure exhaust steam from turbines, special heat transfer fluids or flue gases are also used. 
• Evaporation is differ from drying in that the residue is the liquid, sometimes a highly viscous one rather than solid.
• Evaporation is differ from distillation in that vapor usually is a single component, and even when vapor is a mixture, separation of vapor into a smaller fraction is not carried out in evaporation.

• Single effect Evaporator
• Multiple effect evaporator
• once through and circulation evaporators
• Capacity
• Economy
• Steam consumption
• Heat transfer in Evaporators
• Boiling point elevation
• Boiling point rise
• Comparison of single and multiple effect evaporators 

CLASSIFICATION OF HEAT EXCHANGERS BASED ON CONSTRUCTION

Classification of heat exchangers based on construction

Plate-Type Heat Exchangers

• Plate-type heat exchangers are usually built of thin plates (all prime surface). 
• The plates are either smooth or have some form of corrugation, and they are either flat or wound in an exchanger. 
• Generally, these exchangers cannot accommodate very high pressures,temperatures, or pressure and temperature differences. 

• Plate heat exchangers (PHEs) can be classified as gasketed, welded (one or both fluid passages), or brazed, depending on the leak tightness required. 
• Other plate-type exchangers are spiral plate, lamella, and platecoil exchangers.

Extended Surface Heat Exchangers

• The tubular and plate-type exchangers are all prime surface heat exchangers, except for a shell-and-tube exchanger with low finned tubing. 
• In some applications, much higher (up to about 98%) exchanger effectiveness is essential, and the box volume and mass are limited so that a much more compact surface is mandated.
• Also, in a heat exchanger with gases or some liquids, the heat transfer coefficient is quite low on one or both fluid sides. This results in a large heat transfer surface area requirement.
• One of the most common methods to increase the surface area and exchanger compactness is to add the extended surface (fins) and use fins with the fin density ( fin frequency, fins/m or fins/in.) as high as possible on one or both fluid sides, depending on the design requirement. 
• Addition of fins can increase the surface area by 5 to 12 times the primary surface area in general, depending on the design. 
• The resulting exchanger is referred to as an extended surface exchanger. 
• Flow area is increased by the use of thingauge material and sizing the core properly.
• The heat transfer coefficient on extended surfaces may be higher or lower than that on unfinned surfaces. 

• Plate-fin and tube-fin geometries are the two most common types of extended surface heat exchangers.

Regenerators

• The regenerator is a storage-type heat exchanger. 
• Regenerator, is a type of heat exchanger where heat from the hot fluid is intermittently stored in a thermal storage medium before it is transferred to the cold fluid. 
• To accomplish this the hot fluid is brought into contact with the heat storage medium, then the fluid is displaced with the cold fluid, which absorbs the heat.

• The operation of regenerative heat exchangers is cyclic.
• In the first cycle hot gases/fluids passing through the media heat up the media. In the following cycle, the cold gases pass through the media and they are heated by the already hot media.
• Typical medias used are packed towers of ceramic material with required gaps for the gases to pass through them.
• In regenerative heat exchangers, the fluid on either side of the heat exchanger can be the same fluid. 

CLASSIFICATION OF HEAT EXCHANGER BY CONSTRUCTION

Heat exchangers are frequently characterized by construction features into four major construction types  as follows

1. Tubular heat exchangers
2. Plate-type heat exchangers
3. Extended surface heat exchangers
4. Regenerative heat exchangers

Tubular heat exchanger

• These exchangers are generally built of circular tubes, although elliptical, rectangular, or round/flat twisted tubes.
• Tubular exchangers can be designed for high pressures relative to the environment and high-pressure differences between the fluids.
• Tubular exchangers are used primarily for liquid-to-liquid and liquid-to-phase change (condensing or evaporating) heat transfer applications. 
• They are used for gas-to-liquid and gas-to-gas heat transfer applications primarily when the operating temperature and/or pressure is very high or fouling is a severe problem on at least one fluid side and no other types of exchangers would work. 
• These exchangers may be classified as 

1. Shell-and tube heat exchangers
2. Double-pipe heat exchangers
3. Spiral tube heat exchangers

Shell-and-Tube Exchangers. 

• This exchanger, is generally built of a bundle of round tubes mounted in a cylindrical shell with the tube axis parallel to that of the shell. 
• One fluid flows inside the tubes, the other flows across and along the tubes. 

• A variety of different internal constructions are used in shell-and-tube exchangers, depending on the desired heat transfer and pressure drop performance and the methods employed to reduce thermal stresses, to prevent leakages, to provide for ease of cleaning, to contain operating pressures and temperatures, to control corrosion, to accommodate highly asymmetric flows, and so on. 
• The three most common types of shell-and-tube exchangers are 

1. Fixed tubesheet design
2. U-tube design
3. Floating-head type. 

The major components of this exchanger are tubes (or tube bundle), shell, frontend head, rear-end head, baffles, and tubesheets 

Tubes. 

• Round tubes in various shapes are used in shell-and-tube exchangers. 
• Most common are the tube bundles with straight and U-tubes used exchangers. 
• In most applications, tubes have single walls, but when working with radioactive,reactive, or toxic fluids and potable water, double-wall tubing is used. 
• In most applications, tubes are bare, but when gas or low-heat-transfer coefficient liquid is used on the shell side, low-height fins (low fins) are used on the shell side. 
• Tubes are drawn, extruded, or welded, and they are made from metals, plastics, and ceramics, depending on the applications.

Shells

• The shell is a container for the shell fluid.
• Usually, it is cylindrical in shape with a circular cross section, although shells of different shapes are used. 
• Although the tubes may have single or multiple passes, there is one pass on the shell side. 
• To increase the mean temperature difference and hence exchanger effectiveness, a pure counterflow arrangement is desirable for a two-tube-pass exchanger.  

Nozzles. 

• The entrance and exit ports for the shell and tube fluids, referred to as nozzles
• These are pipes of constant cross section welded to the shell and channels. 
• They are used to distribute or collect the fluid uniformly on the shell and tube sides. 

 Front- and Rear-End Heads. 

• These are used for entrance and exit of the tube fluid.
• The front-end head is stationary, while the rear-end head could be either stationary (allowing for no tube thermal expansion) or floating, depending on the thermal stresses between the tubes and shell. 
• The major criteria for selection of the front-end head are cost, maintenance and inspection, hazard due to mixing of shell and tube fluids, and leakage to ambient and operating pressures. 
• The major criteria for selection of the rear-end head are the allowance for thermal stresses, a provision to remove the tube bundle for cleaning the shell side, prevention of mixing of tube and shell fluids, and sealing any leakage path for the shell fluid to ambient. 

Baffles. 

• Baffles may be classified as transverse and longitudinal types. 
• The purpose of longitudinal baffles is to control the overall flow direction of the shell fluid such that a desired overall flow arrangement of the two fluid streams is achieved.
• Single- and double-segmental baffles are used most frequently due to their ability to assist maximum heat transfer (due to a high-shell-side heat transfer coefficient) for a given pressure drop in a minimum amount of space. 
• Triple and no-tubes-in-window segmental baffles are used for low-pressure-drop applications. 
• The choice of baffle type, spacing, and cut is determined largely by flow rate, desired heat transfer rate, allowable pressure drop, tube support, and flow-induced vibrations. 

Tubesheets. 

• These are used to hold tubes at the ends. 
• A tubesheet is generally a round metal plate with holes drilled through for the desired tube pattern, holes for the tie rods (which are used to space and hold plate baffles), grooves for the gaskets, and bolt holes for flanging to the shell and channel. 
• To prevent leakage of the shell fluid at thetubesheet through a clearance between the tube hole and tube, the tube-to-tubesheet joints are made by many methods, such as expanding the tubes, rolling the tubes, hydraulic expansion of tubes, explosive welding of tubes, stuffing of the joints, or welding or brazing of tubes to the tubesheet. 
• The leak-free tube-to-tubesheet joint made by the conventional rolling process.

Double-Pipe Heat Exchangers

• This exchanger usually consists of two concentric pipes with the inner pipe plain or finned.
• One fluid flows in the inner pipe and the other fluid flows in the annulus between pipes in a counterflow direction for the ideal highest performance for the given surface area

• However, if the application requires an almost constant wall temperature, the fluids may flow in a parallel flow direction. 
• This is perhaps the simplest heat exchanger. 
• Flow distribution is no problem, and cleaning is done very easily by disassembly. 
• This configuration is also suitable where one or both of the fluids is at very high pressure, because containment in the small-diameter pipe or tubing is less costly than containment in a large-diameter shell. 
• Double-pipe exchangers are generally used for small-capacity applications where the total heat transfer surface area required is 50m2 (500 ft2) or less because it is expensive on a cost per unit surface area basis. 
• Stacks of double-pipe or multi tube heat exchangers are also used in some process applications with radial or longitudinal fins. 

Spiral Tube Heat Exchangers. 

• These consist of one or more spirally wound coils fitted in a shell. 
• Heat transfer rate associated with a spiral tube is higher than that for a straight tube.
• In addition, a considerable amount of surface can be accommodated in a given space by spiraling. 
• Thermal expansion is no problem, but cleaning is almost impossible