Henry Darcy and His Law

The History of the Darcy-Weisbach Equation

Glenn Brown
Oklahoma State University
6/7/00
revised 2/21/02

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What we call the Darcy-Weisbach equation has had a long history of development. It is named after two of the great hydraulic engineers of the middle 19th century, but others have also played a major role. Julies Weisbach (1806-1871) a native of Saxony, proposed in 1845 the equation we now use,

h_l = fL/D * V²/2g

where hl is the head loss, L is the pipe length, D is the pipe diameter, V is the average velocity, g is the acceleration of gravity and f is a friction factor. However, he did not provide adequate data for the variation in f with velocity. Thus, his equation performed poorly compared to the empirical Prony equation (Gaspard Clair Francois Marie Riche de Prony, 1755-1839) in wide use at the time;

h_l = L/D * (aV + bV²)

where a and b are empirical friction factors for the velocity and velocity squared.

While Weisbach was ahead of most other engineers, his was not the first work in the area. In about 1770 Antoine Chézy (1718-1798), an early graduate of l’Ecole des Ponts et Chaussées, published an equation for flow in open channels that can be reduced to the same form. Unfortunately, Chézy’s work was lost until 1800 when his former student, Prony published an account describing it. Surprisingly, Prony developed his own equation, but it is believed that Weisbach was aware of Chézy’s work from Prony’s publication. Darcy, (Prony’s student) in 1857 published new relations for the Prony coefficients based on a large number of experiments. His new equation was,

h_l = L/D * [(c + d/D²)V + (d + e/D)V²]

where c, d and e are empirical coefficients for a given type of pipe. Darcy thus introduced the concept of the pipe roughness scaled by the diameter; what we now state as the relative roughness when applying the Moody diagram. Therefore, it is traditional to call f, the “Darcy f factor”, even though Darcy never proposed it in that form. Fanning apparently was the first effectively put together the two concepts in (1877). He published a large compilation of f values as a function of pipe material, diameter and velocity. His data came from French, American, English and German publications, with Darcy being the single biggest source. However, it should be noted that Fanning used hydraulic radius, instead of D in the friction equation, thus “Fanning f” values are only 1/4th of “Darcy f” values.

Parallel to the development in hydraulics, viscosity and laminar flow were defined by Jean Poisseuille (1799-1869) and Gotthilf Hagen (1797-1884), while Osborne Reynolds (1842-1912) described the transition from laminar to turbulent flow in 1883. During the early 20th century, Ludwig Prandtl (1875-1953) and his students Th. von Kármán (1881-1963) Paul Blasius (1883-?) and Johnann Nikuradse (1894-1979) attempted to provide an analytical prediction of the friction factor using both theoretical considerations and data from smooth and uniform sand lined pipes. Their work was complimented by Colebrook and White’s analysis of pipes with non-uniform roughness in 1939. The Darcy-Weisbach equation was not made universally useful until the development of the Moody diagram (Moody, 1944) which built on the work of Hunter Rouse. Rouse (1946) gives a good feel for the development of the f factor, but he doesn’t reference Moody. Rouse felt that Moody was given too much credit for what Rouse himself and others did (Rouse, 1976).

The name of the equation through time is also curious and may be tracked in hydraulic and fluid mechanics textbooks. Early texts generally do not name the equation. Starting in the mid 20th century some authors including at least one German named it “Darcy’s Equation”, an obvious confusion point with “Darcy’s Law”. Rouse in 1946 appears to be the first to call it “Darcy-Weisbach”, but that naming did not become universal until the late 1980’s. It is a good enough name, but as pointed out previously, it leaves out many important contributions. So if you wanted give full credit and confuse people, call it the “Chézy-Weisbach-Darcy-Poiseuille-Hagen-Reynolds-Fanning-Prandtl-Blasius-Kármaán-Nikuradse-Colebrook-White-Rouse-Moody equation”.

From a practical standpoint, the Darcy-Weisbach equation has only become popular since the advent of the electronic calculator. It requires a lot of number crunching compared to empirical relationships, such as the Hazen-Williams equation, which are valid over narrow ranges. However, because of its general accuracy and complete range of application, the Darcy-Weisbach Equation should be considered the standard and the others should be left for the historians. A recent interesting discussion on the topic is presented by Liou (1998), Christensen (2000), Locher (2000) and Swamee (2000).
References

Christensen, B.A., 2000. Discussion of “Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering”, ASCE.

Darcy, H. 1857. Recherches Experimentales Relatives au Mouvement de L’Eau dans les Tuyaux, 2 volumes, Mallet-Bachelier, Paris. 268 pages and atlas. (”Experimental Research Relating to the Movement of Water in Pipes”)

Fanning, 1877. Treatise on Water Supply.

Liou, C.P., 1998. Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering, ASCE. Vol. 124.

Locher, F. A., 2000. Discussion of “Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering”, ASCE.

Moody, L. F., 1944. Friction factors for pipe flow. Transactions of the ASME, Vol. 66.

Poiseuille, J. L., 1841. Recherches expérimentales sur le mouvement des liquides dans les tubes de très-petits diamètres, Comptes Rendus, Académie des Sciences, Paris, 1841.

Reynolds, O., 1883. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channel, Philo. Trans. of the Royal Soc., 174:935-982.

Rouse, H., 1946. Elementary Mechanics of Fluids. John Wiley and Sons, New York.

Rouse, H., 1976. History of Hydraulics in America, 1776-1976.

Swamee, P. K., 2000. Discussion of “Limitations and Proper Use of the Hazen-Williams Equation. Journal of Hydraulic Engineering”, ASCE.

Weisbach, J., 1845. Lehrbuch der Ingenieur- und Maschinen-Mechanik, Braunschwieg.

Source:

http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm

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Unit Operations

• Originally coined as ” unit actions” by Arthur D. Little in 1915, included 12 actions such as pulverizing, mixing, heating, etc.
• A part of a process unit that can be analyzed independently
• Based on the philosophy that the widely varying sequences of steps can be reduced to simple operations or reactions that are identical in fundamentals regardless of the materials being processed

Fundamental Transport Processes

• Momentum Transfer - deals with momentum changes that occur in moving media
• Heat Transfer - deals with transfer of heat
• Mass Transfer - deals with transfer of mass from one phase to another distinct phase

Rate of Transport - is the ratio of driving force and resistance

Driving force

• When two substances or phases not at equilibrium are brought into contact, there is a tendency for a change to take place that will result in an approach toward the equilibrium condition
• It is the difference between the existing condition and the equilibrium condition
• Prime driver in causing change in transport processes

Resistance - impedes the transport operation

Unit Processes

• Also called unit changes or conversion processes
• Commercialization of chemical reactions
• Primarily concerns/involves chemical reactions in the conversion of raw materials to products

Alcoholysis

• In the broadest sense, the addition of hydroxyl group
• Reactions like esterification, formation of ether, etc.

H₂C        CH₂ +  C₂H₅OH  →  C₂H₅OCH₂CH₃OH

O

Alkylene            Ethanol              Hydroxy Ether

Oxide

Alkylation

• Introduction of alkyl group
• In petroleum industries, low molecular weight fractions can be reacted to form higher molecular weight compounds
• Used in the production of anesthetics, antipyretics, dyes, explosives, solvents, plastics etc.

Calcination

• Heating material below melting point resulting to loss of moisture and other volatile compounds
• Materials undergoing such process will also be oxidized or reduced
• Used in treating ores or concentrates, clays etc.

CaCO₃ ^heat CaO  +  CO₂

Combustion

• Burning of any gaseous, liquid or solid substance referred as fuels
• Oxygen source can be either free form as in O₂ or in combination with other elements or compounds like HNO₃, H₂O₂ etc.
• Process often evolves heat and light
• Heat of reaction in combustion is a common source of energy for chemical process industries

Dehydration

• Refers to removal of water of chemical combination from a compound
• Most common way of preparing olefins from alcohols.  Alcohol is passed through alumina (Al₂O₃) catalyst at elevated temperature to remove water.

CH₃CH₂OH      ^{alumina at 673 K} CH₂=CH₂ +  H₂O

Double Decomposition

• Also known as exchange reaction

Electrolysis

• Reactions are carried out in solutions of electrolyte or molten salts by passage of electricity
• Electrodes are immersed in the solution and direct current is passed through the solution
• Oxidation takes places at the positive electrode or anode while reduction takes place in the negative electrode or cathode
• Example, electrolysis of salt solution results to the formation of chlorine gas at the anode while sodium hydroxide and hydrogen gas is formed at the cathode

At anode         Cl₂(g)

NaCl (aq)     ^electricity

At cathode      2 NaOH  + H₂ (g)

Fermentation

• Deals with conversion of one substance to another with the use of microorganisms like yeasts and bacteria
• Used for production of alcohols, acetone, antibiotics, acids like lactic and citric acids, etc.
• Involves complex processes like oxidation, reduction, hydrolysis, esterification, etc.
• Parameters being monitored are temperature, concentration, pH and to a certain extent pressure

Halogenation

• Introduction of halogens
• Most widely used is chlorination due to low cost and usefulness while iodination is seldom used to high cost
• Used for production of insecticides, dye intermediates, industrial chemicals
• Example, reaction of chlorine on methane in the presence of light.  Reaction can not be controlled perfectly resulting to production of several products.

CH₄ +  Cl₂ →  CH₃Cl        +  HCl

CH₃Cl +  Cl₂ →  CH₂Cl₂ +  HCl  and so on

Hydrolysis

• Reaction with water done in the presence of a catalyst
• If reactants are not miscible, emulsifying agents are added
• Examples are saponification of oils and fats, starch conversion, sugar inversion and breaking down of proteins

Ion Exchange

• Refers to the interchange of ions that take place when an ionic solid is contacted with an electrolytic solutions
• Can be used for the
  • Transformation of electrolytes like in water softening where calcium and magnesium ions are exchanged for sodium
  • Removal of ionic constituents like deionization of water
  • Separation of ionic substances like amino acids
  • Concentration of ionic solutions

Neutralization

• Reactions of acids and bases

Nitration

• Generally, refers to treatment of organic compounds with nitric acid to produce either nitrates or nitro compounds

ROH  +  HNO₃ →  RONO₂ +  H₂O

RH     +  HNO₃ →  RNO₂ +  H₂O

• Plays important role as an intermediate reaction
• Hastened by the presence of sulfuric acid
• Used in the manufacture of nitrocellulose (protective coating), explosives, etc

Oxidation (Controlled)

• Oxygen is most often used as oxidizing agents
• In liquid-phase reactions, permanganates, dichromates, chloric anyhydrides, hypochlorites, chlorates, lead peroxide and hydrogen peroxides are used
• Reaction is basically exothermic
• Heat transfer system are carefully designed to prevent controlled oxidation becoming to combustion
• Example, production of phthalic anhydride by air oxidation of naphthalene at 698 K in the presence of V₂O₅ catalyst

Polymerization

• Simple molecules (monomer) react to form larger molecules
• Classified as polymerization through
  • Condensation reactions occur between two groups to form a new group not present in the reactant with a small compound split out like water usually be elimination of water or alcohol from bifunctional molecules
  • Addition by opening of a multiple bond without elimination of any part of molecule

H₂C=CHX  +  H₂C=CHX  →  H₂C-CHXCH2CHX

• Can be carried out in bulk, solution, emulsion or suspension

Pyrolysis

• Unit process where heat is employed to decompose compounds
• Some reactions take place at low temperature in the presence of solvents
• Used in industry to break down large molecules into smaller molecules
• Examples are
  • Wood and coal are heated in a chamber in the absence of air
  • Thermal cracking of heavy hydrocarbon to lighter fractions like gasoline and diesel

Source:

Jose, W.  Introduction to Chemical Engineering. UP Press, 1985.

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