Plate and Frame Heat Exchangers: Implications for Size Reduction

We have seen that alternative technologies have significant size advantage over shell-and-tube heat exchangers. Now let’s consider the implications of this. The first advantage is smaller plot plan for the process plant. The spacing between process equipment can be reduced. So, if the plant is to be housed in a building, the size of the building can be reduced. In any event, the amount of structural steel used to support the plant can be reduced and given the weight saving, the load on that structure is also reduced. The weight advantage extends to the design of the foundations used to support the plant.

Since, the spacing between individual equipment items is reduced, expenditure on piping is reduced. Once more we stress the savings associated with size and weight reduction can only be achieved if these advantages are recognized at the earliest stages of the plant design.

Reduced Plant Complexity

As we will briefly show, the use of alternative exchanger technologies can result in significant reduction in plant complexity. This not only enforces the savings associated with reduced size and weight (reduced plot space, structural cost savings, piping cost reduction etc.) but also has safety implications. The simpler the plant structure the easier it is for the process operator to understand the plant. The simpler the plant structure, the safer, easier and more straight forward the plant maintenance (the fewer the pipe branches that must be blanked etc.).

The alternative technologies result in reduced complexity by reducing the number of heat exchangers. This is achieved through:

• improved ‘thermal contacting’
• multi-streaming.

Mechanical constraints play a significant role in the design of shell-and-tube heat exchangers. For instance, it is common to find that some users place restrictions on the length of the tubes used in such a unit. Such a restriction can have important implications for the design. In the case of exchangers requiring large surface areas the restriction drives the design towards large tube counts. If such tube counts then lead to low tube side velocity, the designer is tempted to increase the number of tube side passes in order to maintain a reasonable tube-side heat transfer coefficient.

Thermal expansion considerations can also lead the designer to opt for multiple tube passes for the cost of a floating head is generally lower than the cost of installing an expansion bellows in the exchanger shell.

The use of multiple tube passes has four detrimental effects. First, it leads to a reduction in the number of tubes that can be accommodated in a given size of shell (so it leads to increased shell diameter and cost). Second, for bundles having more than four tube passes, the pass partition lanes introduced into the bundle give rise to an increase in the quantity of shell-side fluid bypassing the tube bundle and a reduction in tube-side heat transfer coefficient. Thirdly, it gives rise to wasted tube side pressure drop in the return headers. Finally, and most significantly, the use of multiple tube passes results in the thermal contacting of the streams not being pure counter-flow. This has two effects. The first is that the Effective Mean Temperature Driving Force is reduced. The second, and more serious effect, is that a ‘temperature cross’ can occur.

If a ‘temperature cross’ occurs, the designer must split the duty between a number of individual heat exchangers arranged in series. Figures 1 and 2 below illustrate the difference between temperatures that are said to be ‘crossing’ and those that are not.

Many of the alternative heat exchanger technologies allow the application of pure counter-flow across all size and flow ranges. The results are better use of available temperature driving force and the use of single heat exchangers.

Let’s now consider multi-streaming. The traditional shell-and-tube heat exchanger only handles one hot and one cold stream. Some heat exchanger technologies (most notably plate-fin and printed circuit exchangers) can handle many streams. It is not uncommon to find plate-fin heat exchangers transferring heat between ten individual process. Such units can be considered to contain a whole heat exchanger network within the body of a single exchanger. Distribution and recombination of process flows is undertaken inside the exchanger. The result is a major saving in piping cost.

Engineers often over-look the opportunities of using a plate and frame unit as a multi-stream unit. (Again, this will be a regular oversight if exchanger selection is not made until after the flow sheet has been developed).

A good example of multi-streaming is the use of a plate heat exchanger serving as a process interchanger on one side and a trim cooler on the other. This arrangement is particularly useful for product streams that are exiting a process and must be cooled for storage. Another popular function of multi-streaming is in lowering material costs. Often times, once streams are cooled to a certain temperature, they pose much less of a corrosion risk. Half of the exchanger can contain a higher alloy, while the other side can utilize stainless steel or a lower alloy.

In Figure 3 we show how a plate and frame unit has been applied to a problem involving three process streams. The heat transfer properties used for styrene are given in Table 1. Just one unit is used and this unit has 1,335 sq.ft. of effective surface area.

In Figure 4 we show the equivalent shell-and-tube solution. In order to avoid temperature crosses we need six individual exchangers: the cooler having two shells in series (each having 1,440 sq.ft of effective surface); the heat recovery unit having four shells in series (each having 2,116 sq.ft. of surface).

So, our plate-and-frame design involves the use of 1,335 sq.ft. of surface in a single unit. The equivalent shell-and-tube design has 11,344 sq.ft. of surface distributed across four separate exchangers.

Table 1: Heat Transfer Properties Used for Styrene in the Multi-Stream Example

	100 °F	150 °F	200 °F
Density (lb/ft3)	55.5	53.9	52.3
Specific Heat (Btu/lb 0F)	0.427	0.447	0.471
Viscosity (cP)	0.590	0.428	0.329
Thermal Cond. (Btu/ft h 0F)	0.077	0.074	0.070

Data from PhysProps^© by G.P. Engineering, Version 1.5.0

Plate and Frame Heat Exchangers: Implications for Size Reduction

We have seen that alternative technologies have significant size advantage over shell-and-tube heat exchangers. Now let’s consider the implications of this. The first advantage is smaller plot plan for the process plant. The spacing between process equipment can be reduced. So, if the plant is to be housed in a building, the size of the building can be reduced. In any event, the amount of structural steel used to support the plant can be reduced and given the weight saving, the load on that structure is also reduced. The weight advantage extends to the design of the foundations used to support the plant.

Since, the spacing between individual equipment items is reduced, expenditure on piping is reduced. Once more we stress the savings associated with size and weight reduction can only be achieved if these advantages are recognized at the earliest stages of the plant design.

Reduced Plant Complexity

As we will briefly show, the use of alternative exchanger technologies can result in significant reduction in plant complexity. This not only enforces the savings associated with reduced size and weight (reduced plot space, structural cost savings, piping cost reduction etc.) but also has safety implications. The simpler the plant structure the easier it is for the process operator to understand the plant. The simpler the plant structure, the safer, easier and more straight forward the plant maintenance (the fewer the pipe branches that must be blanked etc.).

The alternative technologies result in reduced complexity by reducing the number of heat exchangers. This is achieved through:

• improved ‘thermal contacting’
• multi-streaming.

Mechanical constraints play a significant role in the design of shell-and-tube heat exchangers. For instance, it is common to find that some users place restrictions on the length of the tubes used in such a unit. Such a restriction can have important implications for the design. In the case of exchangers requiring large surface areas the restriction drives the design towards large tube counts. If such tube counts then lead to low tube side velocity, the designer is tempted to increase the number of tube side passes in order to maintain a reasonable tube-side heat transfer coefficient.

Thermal expansion considerations can also lead the designer to opt for multiple tube passes for the cost of a floating head is generally lower than the cost of installing an expansion bellows in the exchanger shell.

The use of multiple tube passes has four detrimental effects. First, it leads to a reduction in the number of tubes that can be accommodated in a given size of shell (so it leads to increased shell diameter and cost). Second, for bundles having more than four tube passes, the pass partition lanes introduced into the bundle give rise to an increase in the quantity of shell-side fluid bypassing the tube bundle and a reduction in tube-side heat transfer coefficient. Thirdly, it gives rise to wasted tube side pressure drop in the return headers. Finally, and most significantly, the use of multiple tube passes results in the thermal contacting of the streams not being pure counter-flow. This has two effects. The first is that the Effective Mean Temperature Driving Force is reduced. The second, and more serious effect, is that a ‘temperature cross’ can occur.

If a ‘temperature cross’ occurs, the designer must split the duty between a number of individual heat exchangers arranged in series. Figures 1 and 2 below illustrate the difference between temperatures that are said to be ‘crossing’ and those that are not.

Many of the alternative heat exchanger technologies allow the application of pure counter-flow across all size and flow ranges. The results are better use of available temperature driving force and the use of single heat exchangers.

Let’s now consider multi-streaming. The traditional shell-and-tube heat exchanger only handles one hot and one cold stream. Some heat exchanger technologies (most notably plate-fin and printed circuit exchangers) can handle many streams. It is not uncommon to find plate-fin heat exchangers transferring heat between ten individual process. Such units can be considered to contain a whole heat exchanger network within the body of a single exchanger. Distribution and recombination of process flows is undertaken inside the exchanger. The result is a major saving in piping cost.

Engineers often over-look the opportunities of using a plate and frame unit as a multi-stream unit. (Again, this will be a regular oversight if exchanger selection is not made until after the flow sheet has been developed).

A good example of multi-streaming is the use of a plate heat exchanger serving as a process interchanger on one side and a trim cooler on the other. This arrangement is particularly useful for product streams that are exiting a process and must be cooled for storage. Another popular function of multi-streaming is in lowering material costs. Often times, once streams are cooled to a certain temperature, they pose much less of a corrosion risk. Half of the exchanger can contain a higher alloy, while the other side can utilize stainless steel or a lower alloy.

In Figure 3 we show how a plate and frame unit has been applied to a problem involving three process streams. The heat transfer properties used for styrene are given in Table 1. Just one unit is used and this unit has 1,335 sq.ft. of effective surface area.

In Figure 4 we show the equivalent shell-and-tube solution. In order to avoid temperature crosses we need six individual exchangers: the cooler having two shells in series (each having 1,440 sq.ft of effective surface); the heat recovery unit having four shells in series (each having 2,116 sq.ft. of surface).

So, our plate-and-frame design involves the use of 1,335 sq.ft. of surface in a single unit. The equivalent shell-and-tube design has 11,344 sq.ft. of surface distributed across four separate exchangers.

Table 1: Heat Transfer Properties Used for Styrene in the Multi-Stream Example

	100 °F	150 °F	200 °F
Density (lb/ft3)	55.5	53.9	52.3
Specific Heat (Btu/lb 0F)	0.427	0.447	0.471
Viscosity (cP)	0.590	0.428	0.329
Thermal Cond. (Btu/ft h 0F)	0.077	0.074	0.070

Data from PhysProps^© by G.P. Engineering, Version 1.5.0

Plate and Frame Heat Exchangers-Preliminary Design, Example

Consider the following example: 150,000 lb/h of water is being cooled from 200 °F to 175 °F by 75,000 lb/h of SAE 30 oil. The oil enters the exchanger at 60 °F and leaves at 168 °F. The average viscosity of the water passing through the unit is 0.33 cP and the average viscosity of the oil in the unit is 215 cP. The maximum allowable pressure drop through the plate heat exchanger is 15 psig on the hot and cold sides

Step 1 : Calculate the LMTD

Step 2: Calculate NTU_HOT and NTU_COLD

Step 3: Read h_Hot from 0.25 < NTU < 2.0 chart for hydrocarbons

Although is there not a viscosity line for 215 cP, the line representing “100 cP” can be or viscosities up to about 400-500 cP. The heat exchanger will be pressure drop limited and the heat transfer coefficient will not change appreciably over this viscosity range for plate and frame exchangers. Reading from the chart, a pressure drop of 15 psig corresponds to h_Hot @ 50 Btu/h ft2 °F

Step 4: Read h_Cold from 0.25 < NTU < 2.0 chart for water based liquids

Again, you will note that the exact viscosity line needed for pure water (0.33 cP) in this case is not available. However, the “1.0 cP” line on the chart will provide a very good estimate of the heat transfer coefficient that pure water will exhibit. Reading from the chart, a pressure drop of 15 psig corresponds to h_Cold @ 3000 Btu/h ft2 °F

Step 5: Calculate the Overall Heat Transfer Coefficient (OHTC)

Assume a stainless steel plate with a thickness of 0.50 mm is being used. 316 stainless steel has a thermal conductivity of 8.67 Btu/h ft °F.

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150,000 lb/h of water is being cooled from 200 °F by 150,000 lb/h of NaCl brine. The brine enters the exchanger at 50 °F and leaves at 171 °F. The average viscosity of the water passing through the unit is 0.46 cP and the average viscosity of the brine in the unit is 1.10 cP. The maximum allowable pressure drop through the plate heat exchanger is 10 psig on the hot (water) side and 20 psig on the cold (brine) side.

As before, the LMTD is calculated to be 38.5 °F. NTU_Hot and NTU_Cold are calculated as 2.59 and 3.14 respectively. Reading h_Hot and h_Cold from the chart for 2.0 < NTU < 4.0 (water based), gives about 2000 Btu/h ft2 °F and 2500 Btu/h ft2 °F respectively. Although the material of choice may be Titanium or Palladium stabilized Titanium, we will use the properties for stainless steel for our preliminary sizing. Calculating the OHTC as before yields 918 Btu/h ft2 °F.

Plate and Frame Heat Exchangers: Preliminary Design

A Quick Look at the Basics

Numerous articles have been published regarding the advantages of compact heat exchangers. Briefly, their higher heat transfer coefficients, compact size, ease of service, cost effectiveness, and their unique ability to handle fouling fluids make compact exchangers a good choice for many services.

Plate heat exchangers consist of pressed, corrugated metal plates fitted between a thick, carbon steel frame. Each plate flow channel is sealed with a gasket, a weld, or an alternating combination of the two. It is not uncommon for plate and frame heat exchangers to have overall heat transfer coefficient that are 3-4 times those found in shell and tube heat exchangers.

Specifying Plate and Frame Heat Exchangers

Engineers often fail to realize the differences between heat transfer technologies when preparing a specification. This specification is then sent to vendors of different types of heat exchangers. Consider the following example:

A process stream requires C276 material to guard against corrosion. The stream needs to be cooled with cooling water before being sent to storage. The metallurgy makes the process stream an immediate candidate for the tubeside of a shell and tube heat exchanger. The cooling water is available at 80 °F and must be returned at a temperature no higher than 115 °F. The process engineer realizes that with the water flow being placed on the shellside, larger flowrates will enhance the heat transfer coefficient. The basis for the heat exchanger quotation was specified as follows:

	Tubeside	Shellside
Flow rate (GPM)	500	1800
Temperature In (°F)	280	80
Temperature Out (°F)	150	92
Allowable Pressure Drop (psig)	15	15

According to the engineer’s calculations, these basic parameters should provide a good shell and tube design with a minimum amount of C276 material (an expensive alloy). The completed specification sheet is forwarded to many manufacturers, including those that could easily quote plate and frame or another compact technology. A typical plate and frame unit designed to meet this specification would have about 650 ft² of area compared to about 420 ft² for a shell and tube exchanger. A plate and frame unit designed to the above specification is limited by the allowable pressure drop on the cooling water. If the cooling water flow is reduced to 655 GPM and the outlet water temperature allowed to rise to 115 °F, the plate and frame heat exchanger would contain about 185 ft² of area. The unit is smaller, less expensive, and uses less water. The load being transferred to the cooling tower is the same.

The theory that applied to the shell and tube heat exchanger (increasing water flow will minimize heat transfer area), works in exactly the opposite direction for compact technologies. The larger water flow actually drives the cost of the unit upward. Rather than supplying a rigid specification to all heat exchanger manufacturers, the engineer should have explained his goal in regards to the process stream. Then he could have stated the following:

“The process stream is to be cooled with cooling water. Up to 2000 GPM of water is available at 80 °F. The maximum return temperature is 115 °F.”

This simple statement could result in vastly different configurations when compared with the designs that would result from the original specification.

Design Charts for Plate and Frame Exchangers (Download Printable Copies in MS Excel)

Often, in compact heat transfer technology, engineers find themselves at the mercy of the manufacturers of the equipment. For example, limited literature correlations are available to help in the preliminary design of plate and frame heat exchangers. We will introduce a series of charts that can be used for performing preliminary sizing of plate and frame exchangers. After introducing the charts, we will follow with examples to help clarify the use of the charts. The following should be noted regarding the use of the charts:

1. These charts are valid for single pass units with 0.50 mm thick plates. The accuracy of the charts will not be compromised for most materials of construction.
2. Wetted material thermal conductivity is taken as 8.67 Btu/(h ft °F) (value for SS)
3. Heat transfer correlations are valid for single phase, liquid-liquid designs
4. The following physical properties were used for the basis:

   	Hydrocarbon-based fluids	Water-based fluids
   Thermal Conductivity (Btu/h ft °F)	0.06	0.33
   Density (lb/ft³)	55	62
   Heat Capacity (Btu/lb 0F)	0.85	0.85

5. Degree of accuracy should be within ± 15% of the service value for the overall heat transfer coefficient, assuming a nominal 10% excess heat transfer area.
6. For fluids with viscosities between 100 and 500 cP, used the 100 cP line of the graphs. For fluids in excess of 500 cP, consult with manufacturers.

Charts

Heuristics for Chemical Process Synthesis

A strategy is recommended that involves assembling the process operations in a specific order, as follows:

Chemical reactions (to eliminate differences in molecular type)

Mixing and recycle (to distribute the chemicals)

Separations (to eliminate differences in composition)

Temperature, pressure, and phase change

Task intregation (to combine operations into unit process) The heuristics/the rules of thumb in that strategy are: Reaction Operations

Select raw material and chemical reactions to avoid, or reduce, the handling and storage of hazardous and toxic chemcial. Distribution of chemicals

Use an excess of one chemical reactant in a reaction operation to consume completely a valuable, toxic, or hazardous chemical reactant.

When nearly pure product are required, eliminate inert species before the reaction operation when the separations are easily accomplished and when the catalist is adversely affected by the inert, but not when a large exothermic heat of reaction must be removed.

Introduce purge stream to provide exits for species that enter the process as impurities in the feed or are formed in irreversible side reactions, when these species are in trace quantities and/or are difficult to separate from the other chemical. Lighter species leave in vapor purge streams, and heavier in the liquid streams.

Do not purge valuable species or species that toxic and hazardous, even in small concentrations. Add separators to recover valuable species. Add reactors to eliminate the hazardous and toxic species.

By-product that produced in reversible reactions, in small quantities, are usually not recovered in separators or purged. Instead, they are usually recycled to extinction.

For competing reactions, both in series and parallel, adjust the temperature, pressure, and catalyst to obtain high yields of the desired products. In the initial distribution of chemical, assume that these conditions can be satisfied. Before developing a base-case design, obtain kinetics data and check this assumption.

For reversible reactions especially, consider conducting them in separation device capable of removing the products, and hence driving the reactions to the right. Such reaction separation operations lead to very different distributions of chemicals. Separations operations

Separate liquid mixtures using ditillation, stripping towers, enhanced (extractive, azeotropic, reactive) distillation, liquid-liquid extraction, crystallizers and/or absorbers.

Attempt to condense vapor mixtures with cooling water. Then use heuristic point 9 (before this point)

Separate vapor mixtures using partial condensers, cryogenic distilliation, absorbtion towers, adsorbers, and/or membrane devices. Heat removal and addition

To remove a highly exothermic heat of reaction, consider the use of excess reactant, an inert diluent, or cold shots. These affect the distribution of chemicals and should be inserted early in process synthesis.

For less exothermic heat of reaction, circulate reactor fluid to and external cooler, or use a jacketed vessel or cooling coils. Also, consider the use of intercoolers between adiabatic reaction stages.

To control temperature for highly endothermic heat of reaction, consider the use of excess reactant, an inert diluent, or hot shots. These affect the distribution of chemicals and should be inserted early in the process synthesis.

For less endothermic heat of reaction, circulate reactor fluid to an external heater, or use a jacketed vessel or heating coils. Also, consider the use of interheaters between the adiabatic reaction stages. Pressure-change operations

To increase the pressure of a stream, pump a liquid rather than compress a gas; that is, condense a vapor, as long as refrigeration (and compression) is not needed, before pumping.

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Source : Seider, W.D., Seader, J.D., and Lewin, D.R., PROCESS DESIGN PRINCIPLE, Synthesis, Analysis, and Evaluation, John Wiley and Sons, Inc., p. 133, 1999.

Determination of Shell and Tube Heat Exchanger Shell Diameter

For triangular pitch proceed as follows:

Draw the equilateral triangle connecting three adjacent tube centers. Any side of the triangle is the tube pitch (recall 1.25 D_o is minimum).

Triangle area is 0.5bh where b is the base and h is the height.

This area contains 1/2 tube.

Calculate area occupied by all the tubes.

Calculate shell diameter to contain this area.

Add one tube diameter all the way around (two tube diameters added to the diameter calculated above).

The result is minimum shell diameter. There is no firm standard for shell diameter increments. Use 2-inch increments for initial planning.For square pitch proceed similarly.

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Source : Branan, C. R., The Process Erzgineer- S Pocket Handbook,Vol. 1, Gulf Publishing Co., p. 54, 1976.

Selection Guides for Heat Exchangers

Type Designation	Significant Feature	Applications Best Suited	Limitations	Relative Cost in Carbon Steel Construction
Fixed Tube Sheet	Both tube sheets fixed to shell	Condensers; liquid-liquid; gas-gas; gas-liquid; cooling and heating, horizontal or vertical, reboiling	Temperature difference at extremes of about 200°F. Due to differential expansion	1.0
Floating Head or Tube Sheet (Removable and nonremovable bundles)	One tube sheet “floats” in shell or with shell, tube bundle may or may not be removable from shell, but back cover can be removed to expose tube ends.	High temperature differentials, above about 200°F. extremes; dirty fluids requiring cleaning of inside as well as outside of shell, horizontal or vertical.	Internal gaskets offer danger of leaking. Corrosiveness of fluids on shell side floating parts. Usually confined to horizontal units.	1.28
U-Tube; U-Bundle	Only one tube sheet required. Tubes bent in Ushape. Bundle is removable.	High temperature differentials which might require provision for expansion in fixed tube units. Clean service or easily cleaned conditions on both tube side and shell side. Horizontal or vertical.	Bends must be carefully made or mechanical damage and danger of rupture can result. Tube side velocities can cause erosion of inside of bends. Fluid should be free of suspended particles.	1.08
Kettle	Tube bundle removable as U-type or floating head. Shell enlarged to allow boiling and vapor disengaging.	Boiling fluid on shell side, as refrigerant, or process fluid being vaporized. Chilling or cooling of tube side fluid in refrigerant evaporation on shell side.	For horizontal installation. Physically large for other applications.	1.2-1.4
Double Pipe	Each tube has own shell forming annular space for shell side fluid. Usually use externally finned tube.	Relatively small transfer area service, or in banks for larger applications. Especially suited for high pressures in tube above 400 psig.	Services suitable for finned tube. Piping-up a large number often requires cost and space.	0.8-1.4
Pipe Coil	Pipe coil for submersion in coil-box of water or sprayed with water is simplest type of exchanger.	Condensing, or relatively low heat loads on sensible transfer.	Transfer coefficient is low, requires relatively large space if heat load is high.	0.5-0.7
Open Tube Sections (Water cooled)	Tubes require no shell, only end headers, usually long, water sprays over surface, sheds scales on outside tubes by expansion and contraction. Can also be used in water box.	Condensing, relatively low heat loads on sensible transfer.	Transfer coefficient is low, takes up less space than pipe coil.	0.8-1.1
Open Tube Sections (Air Cooled) Plain or finned tubes	No shell required, only end heaters similar to water units.	Condensing, high level heat transfer.	Transfer coefficient is low, if natural convection circulation, but is improved with forced air flow across tubes.	0.8-1.8
Plate and Frame	Composed of metal-formed thin plates separated by gaskets. Compact, easy to clean.	Viscous fluids, corrosive fluids slurries, High heat transfer.	Not well suited for boiling or condensing; limit 350-500°F by gaskets. Used for Liquid-Liquid only; not gas-gas.	0.8-1.5
Spiral	Compact, concentric plates; no bypassing, highturbulence.	Cross-flow, condensing, heating.	Process corrosion, suspended materials.	0.8-1.5
Small-tube Teflon	Chemical resistance of tubes: no tube fouling.	Clean fluids, condensing, cross-exchange.	Low heat transfer coefficient.	2.0-4.0

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Source :

Ludwig, E. E., Applied Process Design for Chemical and Petrochemical Plants, 2nd Ed., Vol. 3, Gulf Publishing Co., 1983.

GPSA Engineering Data Book, Gas Processors Suppliers Association, 10th Ed., 1987.

Piping Pressure Drop

A handy relationship for turbulent flow in commercial steel pipes is:

DP_F = W^1.8µ^0.2/20.000d^4.8r

where:

DPF	=	Frictional pressure loss, psi/lOO equivalent ft of Pipe
W	=	Flow rate, lb/hr
µ	=	Viscosity, cp
r	=	Density, lb/ft³
d	=	Internal pipe diameter, in.

This relationship holds for a Reynolds number range of 2,100 to 10⁶. For smooth tubes (assumed for heat exchanger tubeside pressure drop calculations), a constant of 23,000 should be used instead of 20,000.

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Source : Branan, Carl R. "Estimating Pressure Drop," Clzenzicnl Engineel-irzg, August 28. 1978.

Velocity Head

Two of the most useful and basic equations are

Dh = u²/(2g)
DP(V) + Du²/(2g) + DZ + E = 0

where :

Dh	=	Head loss in feet of flowing fluid
u	=	Velocity in ft/sec
g	=	32.2 ft/sec2
P	=	Pressure, lb/ft2
V	=	Specific volume, ft3/lb
Z	=	Elevation, feet
E	=	Head loss due to friction in feet of flowing fluid

In Equation 1 Dh is called the “velocity head.” This expression has a wide range of utility not appreciated by many. It is used “as is” for

1. Sizing the holes in a sparger
2. Calculating leakage through a small hole
3. Sizing a restriction orifice
4. Calculating the flow with a pitot tube

With a coefficient it is used for

1. Orifice calculations
2. Relating fitting losses, etc.

For a sparger consisting of a large pipe having small holes drilled along its length Equation 1 applies directly. This is because the hole diameter and the length of fluid travel passing through the hole are similar dimensions. An orifice on the other hand needs a coefficient in Equation 1 because hole diameter is a much larger dimension than length of travel (say ¹/₈ in. for many orifices).

For compressible fluids one must be careful that when sonic or “choking” velocity is reached, further decreases in downstream pressure do not produce additional flow. This occurs at an upstream to downstream absolute pressure ratio of about 2 : 1. Critical flow due to sonic velocity has practically no application to liquids. The speed of sound in liquids is very high.

Still more mileage can be gotten out of Dh = u²/2g when using it with Equation 2, which is the famous Bernoulli equation. The terms are

1. The PV change
2. The kinetic energy change or “velocity head”
3. The elevation change
4. The friction loss

These contribute to the flowing head loss in a pipe. However, there are many situations where by chance, or on purpose, u²/2g head is converted to PV or vice versa.

We purposely change u2/2g to PV gradually in the following situations:

1. Entering phase separator drums to cut down turbulence and promote separation
2. Entering vacuum condensers to cut down pressure drop

We build up PV and convert it in a controlled manner to u²/2g in a form of tank blender.

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Source :
Branan, C. R. The Process Engineer’s Pocket Handbook, Vol. 1, Gulf Publishing Co., Houston, Texas, p. 1, 1976.

A Mistake in Kern's Heat Transfer

Chemical engineers must be familiar with this set of equations, for designing a shell & tube heat exchanger.
The third equation for F_T then plotted in some figures, corresponded to type of shell & tube exchanger flow. for example in the figure below. This set can be found in any reference that talk about shell & tube heat exchanger, and I think all of that refer to the main source, Kern's "Heat Transfer".

Let's check these cases out...

< case 1 >A stream 90 °C is going to be cooled to 70 °C using cold stream 30 °C, which its temperature will rise to 50 °C. We'll find, the value of R will be 1. Use this value in the equation for F_T & LMTD. The value will be indefinite. So we cannot calculate the first equation.
< MyOpinion >The temperature differences are equal in both sides (T1-t2 = T2-t1), thus we can assume that the weighted/average temperature difference in this system is the same of those temperature differences, so we could use one of those. We can replace the F_T.LMTD in the first equation with TD, which TD=T1-t2=T2-t1. We can see the similar condition in the system which use latent heat in both streams.

< case 2 >A stream 90 °C is going to be cooled to 60 °C using cold stream 30 °C, which its temperature will rise to 70 °C. In this system, we can calculate the value of LMTD, but not for the F_T (it will be indefinite), so we cannot calculate the first equation for Q.
< MyOpinion >Examine the system, we will find that there will be cross temperature difference there. So, we cannot apply the combination of parallel-countercurrent (1-2, 1-4, 1-6, 1-8) to this system. We must use the fully countercurrent (with 1-1 exchanger) to this system, and replace the F_T.LMTD with LMTD only, in the first equation.

Those are my opinions. I hope there will be other opinions, or comments. I'm affraid, perhaps i missunderstood the problem, then please send comment, or tell me the truth....thanks :)