This report reconciles the numerical results of both forms of the Colebrook-White Friction Factor equations. It should
be noted that although the Colebrook equation represents the data to within +/- 10 percent, the equations are not
reconciled to within the 5 decimal place. This reconciliation is of great importance to the researches of explicit-forms
of the Colebrook equation as it now can be compared to
either form-1 (Used in the generation of the Moody Diagram) or form-2, preferred by some investigators. The author prefers form-1 as it is the most compact-form
This presentation is a brief descriptive procedure of simulating in aspen flare system analyser (otherwise called as flarenet). It gives a step by step instructions to develop a flare network scheme in the simulator
Natural gas condensates can form liquid slugs in transmission lines. This presentation describes alternative modelling strategies to determine slug volumes
AFS slug catcher sizing in a gas gathering systemAdam Murray
Elijah Kempton of AFS gave a great lunch and learn to go over slug catcher sizing in a gas gathering system. He discusses pigging and other considerations when engineering your gathering system.
The presentation is for the engineers of HIRA POWER PLANT,. The complete calculations for calculation of boiler efficiency are described in the presentation
This presentation is a brief descriptive procedure of simulating in aspen flare system analyser (otherwise called as flarenet). It gives a step by step instructions to develop a flare network scheme in the simulator
Natural gas condensates can form liquid slugs in transmission lines. This presentation describes alternative modelling strategies to determine slug volumes
AFS slug catcher sizing in a gas gathering systemAdam Murray
Elijah Kempton of AFS gave a great lunch and learn to go over slug catcher sizing in a gas gathering system. He discusses pigging and other considerations when engineering your gathering system.
The presentation is for the engineers of HIRA POWER PLANT,. The complete calculations for calculation of boiler efficiency are described in the presentation
In this Webinar we explore the details of actual case studies focusing on various signs that indicate if a particular support needs replaced or adjusted. The presentation also covers projects that brought on new standard designs and custom designs for specific applications. And lastly, it includes valuable additions and procedures that extend the typical life span of a pipe support. PT&P's field services division is available 24x7 in the case of an emergency situation or for a scheduled shut down.
In this Webinar we explore the details of actual case studies focusing on various signs that indicate if a particular support needs replaced or adjusted. The presentation also covers projects that brought on new standard designs and custom designs for specific applications. And lastly, it includes valuable additions and procedures that extend the typical life span of a pipe support. PT&P's field services division is available 24x7 in the case of an emergency situation or for a scheduled shut down.
The paper presents an analytical solution for the seepage problem under overflow spillway and
founded on complex formations. The spillway was furnished with downstream blanket and two end sheetpiles
to avoid the uplift and piping effects. All investigated models were founded on anisotropic foundation where
the coefficient of permeability in horizontal direction more than that of vertical direction. The analytical
solution was based on the fragment methodology. The study led to a criterion for designing the blanket and
investigating the optimal length of downstream blanket. Also, the effect of the anisotropic soil foundation was
indicated. The simple and practical design charts are presented.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Bound State Solution to Schrodinger Equation with Hulthen Plus Exponential Co...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger with Hulthen plus
exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvarov
method. We obtained both the eigen energy and the wave functions to non -relativistic wave equations. We
implement Matlab algorithm to obtained the numerical bound state energies for various values of
adjustable screening parameter at various quantum state.. The developed potential reduces to Hulthen
potential and the numerical bound state energy conform to that of existing literature.
Math cad ROR solution using a biquadratic bypass methodJulio Banks
This file shows the solution of a ROR (Rate Of Return) using a symbolic solution of a
biquadratic bypass method. The solution is given in terms of a variable, w, appearing
twice in a polynomial of the form. w raised to the 2p, and w raised to the p with a
nontrivial constant. Letting x = w raised to the pth-power as a bypass variable, then a
quadratic bypass equation is solved for x. Since the original root is w, then one must
perform a post-process of x to transform the x-root to the w-root solution sought. It should be noted that the intermediate bypass parameter, w, was eliminated from the
numerical solution.
Conclusion
The objective of this report is to promote the recognition of biquadratic equations
and therefore eliminate the unnecessary iterative procedure otherwise required in
the absence of a closed-form root finding solution. This goal has been achieved in
this report via the utilization of the biquadratic equation solution method.
Apologia - A Call for a Reformation of Christian Protestants Organizations.pdfJulio Banks
This document shows how to know whether an organization claiming to be IRS 501(C)(3) tax exempted nonprofit is being partisan by teaching that the republican party is the party of Jesus Christ violating the nonpartisan IRS requirement are false Christian organizations.
The treatment of large structural systems may be simplified by dividing the system into
smaller systems called components. The components are related through the
displacement, and force conditions at their junction points. Each component is represented
by mode shapes (or functions).
The synchronicity time or common time of two (2) independent and asynchronous events can be readily
completed in a four (4) steps algorithm to be described in this article. Additionally, two (2) illustrative
examples are also provided for completeness of presentation of the Synchronicity Algorithm.
Math cad prime the relationship between the cubit, meter, pi and the golden...Julio Banks
It has been reported that the ancient Egyptians knew pi, the golden ration (phi) and the meter. This paper summarizes the relationship of pi, and phi via the cubit.
Jannaf 10 1986 paper by julio c. banks, et. al.-ballistic performance of lpg ...Julio Banks
This paper is the result of the response of LPG (Liquid Propellant Gun) to high-temperate simulated-desert condition using ANSYS FEA (Finite Element Analysis) and Mica Heating Banks.
web site: http://www.joycenter.net/wp-content/uploads/2013/04/Mans-Search-for-Meaning-Viktor-Frankl.pdf
A case can be made that since the main basis of "The Theory" of evolution is the "Self-preservation principle". That is, how could the propagation of the a specie be enhanced by the demeaning action of a group against its constituents and even self-against-self. The only explanation is that humas were created and not a result of a random sett of actions causing consciousness arriving from non-conscious matter. Life comes from life, and intelligence (DNA), comes from intelligence. This book can be contrasted with: The Lucifer Effect Understanding How Good People Turn Evil by Philip Zimbardo' and also with the Bible for a view of The Meaning of Life from ancient to contemporary writings for balance understanding of the physical (Psyche) to the metaphysical (Spiritual). We can view the human condition as the effect of gravity of interacting physical objects and human interaction as the response to spiritual influence (angels and demons).
When the Scriptures command us to “love your enemies”, “bless those who curse you” as well as “Judge not because as you judge you will also be judged and the very same standard you apply in condemning others you also be condemned”. One can see, in the context of this essay, that we are actually harming ourselves if we do not love our “perceived enemies” since it is well known that “There are no greater flaws than the ones we perceive in others than the ones residing within ourselves (known as “our personal demons” or “small judging sub-self)”. This is in essence the basis of psychological projection which is a form of psychological dissonance (the closer one is to danger, the more one tends to minimize or even ignore its potential harm to us). Psychological dissonance is a defense mechanism by which “the human mind minimizes the pain of being less than the person that we all believe we are and we seek to find. The true self is the enlightened being who has discovered the meaning of love. Finally, we are home when we feel at peace at being who we are; unique individuals just as our DNA and fingerprints identifies us from billions of other humans. Be yourself and celebrate your uniqueness then you will be truly enlightened - you have arrived at the place call true self and therefore happy
The first step required to defeating an enemy is by first thoroughly defining it. A physician runs tests of their patients to determine the type of pathogen ailing such patients. Similarly, we must be clear that A Muslim Terrorist is indeed a Fundamentalist and not a Radical Terrorist since the method of striking terror is explicitly and clearly defined in the Qur'an. Christianity is the only religion that needs not attack any other religion such as the Islam religion since God is the author and finisher of our faith and also commands the Christians to allow God to avenge Himself for our attacks even from Islamic Terrorist. A Christian is commanded to live at peace with all humans and when such a peaceful coexistence is not achieved then we must simply stay away from such toxic humans.
The primary test for a true religion is that "The Judeo-Christian God does not need nor require that mere mortal human beings to defend Him".
NASA-TM-X-74335 --U.S. Standard Atmosphere 1976Julio Banks
NASA-TM-X-74335 --U.S. Standard Atmosphere 1976. This information is useful for airframes (e.g., missiles and aircrafts) aero-thermos analysis and design.
Mathcad P-elements linear versus nonlinear stress 2014-t6Julio Banks
This work couples the classical ESED (Equivalent Strain Energy Density) Method; aka, Glinka. The most expedient method of solving a structural problem using FEA (Finite Element Analysis). There would be occasions when stress concentrations would be calculated due to interior corners, holes, sudden change of geometry (aka stress raisers). Although some software would allow regions in the vicinity of such stress risers to be defined by nonlinear material models such as "Elastic Perfectly-plastic", "Bilinear (Elastic and linear plastic), or the fundamental Ramberg-Osgood metal strain-stress models. Once the Linear-elastic FEA solution is obtained one can readily determine that Pseduo nonlinear strain, the corresponding stress and the implicit stress-intensification factor, Kt. It should be noted that once the analyst-designer is ready for final analysis, it would be most prudent to create a FEA model in which the regions of high concentration of stress to be modeled with local nonlinear models of the metal using St. Vennants' Principle of load-and-resistance distance from area of interest. The P-method is an excellent FEA element that can "find the actual nonlinear stress" by the simple iterative increase of the order of the polynomial representing the stress fields within every P-element. It should be noted that this research was facilitated by the use of the P-element FEA software called StressCheck which is 100% P-element solution which I am quite pleased to have had the opportunity of utilizing for this research.
Apologia - The martyrs killed for clarifying the bibleJulio Banks
I know all the things you do, that you are neither hot nor cold. I wish that you were one or the other! 16But since you are like lukewarm water, neither hot nor cold, I will spit you out of my mouth!” - Revelation 3:15-16
“A man who does not have something for which he is willing to die is not fit to live.” - Martin Luther King Jr.
Spontaneous creation of the universe ex nihil by maya lincoln and avi wasserJulio Banks
Questions regarding the formation of the Universe and ‘what was there ’ before it came to existence have
been of great interest to mankind at all times. Several suggestions have been presented during the ages –
mostly assuming a preliminary state prior to creation. Nevertheless, theories that require initial conditions
are not considered complete, since they lack an explanation of what created such conditions. We therefore
propose the ‘Creatio Ex Nihilo ’ (CEN) theory, aimed at describing the origin of the Universe from ‘nothing ’ in
information terms. The suggested framework does not require amendments to the laws of physics: but rather
provides a new scenario to the Universe initiation process, and from that point merges with state-of-the-art
cosmological models. The paper is aimed at providing a first step towards a more complete model of the
Universe creation – proving that creation Ex Nihilo is feasible. Further adjustments, elaborations, formalisms
and experiments are required to formulate and support the theory.
The “necessary observer” that quantum mechanics require is described in the b...Julio Banks
This essay is intended to share the vies of the author of his Judeo-Christian belief and the physical validation of such believes based upon the theories of Quantum Mechanics.
A fund way to remember how to "fix our manifested creation" by means of observation is as follows: "Keep an eye on the ball", "Do not drop the ball"
Advances in fatigue and fracture mechanics by grzegorz (greg) glinkaJulio Banks
Professor Grzegorz (Greg) Glinka has made substantial contributions to the field of stress concentration evaluation using linear FEA results using the ESED (Equivalent Striain Energy Density). ESED aka Glinka methods allows the determination of strain-stress state at a point of local concentration by equating the strain energy from the linear FEA area in the material strain-stress curve to that of the actual strain-stress of the material using a models such as Ramberg-Osgood. The ESED method is more accurate than the Neuber requiring the equating of SED (Strain Energy Densities) of linear FEA results that Stress is proportional to strain even when the FEA predicts a stress greater than the ultimate strength of the material. One easy method of remember when to use ESED versus Neuber is that ESED, more accurate, should be use on the stress analysis of rocket structures and Neuber delegated to aerospace engines and components.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
1. Fluid Flow 2.9
For fully developed laminar-viscous flow in a pipe, the loss is
evaluated from Equation (8) as follows:
(27)
where Thus, for laminar flow, the
friction factor varies inversely with the Reynolds number.
With turbulent flow, friction loss depends not only on flow con-
ditions, as characterized by the Reynolds number, but also on the
nature of the conduit wall surface. For smooth conduit walls, empir-
ical correlations give
(28a)
(28b)
Generally, f also depends on the wall roughness ε. The variation
is complex and best expressed in chart form (Moody 1944) as
shown in Figure 13. Inspection indicates that, for high Reynolds
numbers and relative roughness, the friction factor becomes inde-
pendent of the Reynolds number in a fully-rough flow regime. Then
(29a)
Values of f between the values for smooth tubes and those for the
fully-rough regime are represented by Colebrook’s natural rough-
ness function:
(29b)
A transition region appears in Figure 13 for Reynolds numbers
between 2000 and 10 000. Below this critical condition, for smooth
walls, Equation (27) is used to determine f ; above the critical con-
dition, Equation (28b) is used. For rough walls, Figure 13 or Equa-
tion (29b) must be used to assess the friction factor in turbulent flow.
To do this, the roughness height ε, which may increase with conduit
use or aging, must be evaluated from the conduit surface (Table 2).
Fig. 13 Relation Between Friction Factor and Reynolds Number
(Moody 1944)
HL( )f
L
ρg
------
8µV
R
2
----------
32LνV
D
2
g
-----------------
64
VD ν⁄
---------------
L
D
----
V
2
2g
------
= = =
Re VD ν and f 64 Re.⁄=⁄=
f
0.3164
Re
0.25
----------------= for Re 10
5
<
f 0.0032
0.221
Re
0.237
-----------------+= for 10
5
Re 3< < 10
6
×
1
f
--------- 1.14 2 log D ε⁄( )+=
1
f
--------- 1.14 2 log D ε⁄( )+= 2 log 1
9.3
Re ε D⁄( ) f
--------------------------------+–
The Reconciliation of the Colebrook-White Friction Factor Equations
by
Julio C. Banks, MSME, PE
e-Mail: Sell-A-Vision@Outlook.com
2001_Fundamentals_Handbook.pdf
2. Colebrook-White Friction Factor Equation Derivation 1 of 3.mcdx Page 1 of 2
Reference: "ASHRAE HVAC 2001 Fundamental Handbook" Equation 29b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
D
ε
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎜
⎝
+1 ――――
9.311
⋅⋅Re ―
ε
D
‾‾f
⎞
⎟
⎟
⎠
((29 b))
The constants 1.138 and 9.311 are results of the reconciliation by the author of the
contributing equations. Transform Equation 29b to equation 2
=λ ―
ε
D
((1))
=――
1
‾f
⋅−2 log
⎛
⎜
⎜
⎝
+―
λ
a
0
―――
b
0
⋅Re ‾f
⎞
⎟
⎟
⎠
((2))
Symbolic Solution
Substitute Eq. 1 into Eq. 29 b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
1
λ
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎝
+1 ――――
9.311
⋅⋅Re λ ‾‾f
⎞
⎟
⎠
((3))
=――
1
‾f
−(( −1.138 ⋅2 log((λ)))) ⋅2 log
⎛
⎜
⎝
⋅
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
―
1
λ
⎞
⎟
⎠
=――
1
‾f
−1.138 ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((4))
Let =⋅−2 log((x)) 1.138 ((5))
Solve for x
≔x =10
⎛
⎜⎝
−――
1.138
2
⎞
⎟⎠
0.2698 ((6))
Substitute Eq. 5 into Eq. 4
=――
1
‾f
−⋅−2 log((x)) ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((7))
Julio C. Banks, PE
3. Colebrook-White Friction Factor Equation Derivation 1 of 3.mcdx Page 2 of 2
=――
1
‾f
⋅−2 log
⎛
⎜
⎝
+⋅λ x ―――
⋅9.311 x
⋅Re ‾‾f
⎞
⎟
⎠
((8))
Recall Equaton 2
=――
1
‾f
⋅−2 log
⎛
⎜
⎜
⎝
+―
λ
a
0
―――
b
0
⋅Re ‾f
⎞
⎟
⎟
⎠
((2))
Compearing equations 8 and 2 provides for the numerical balues of the constant
parameters, and .a
0
b
0
≔a
0
=―
1
x
3.707 ((9))
and
≔b
0
=⋅9.311 x 2.512 ((10))
The standard Colebrook-White equation (used in the generation of the Moody Diagram) is
=――
1
‾f
⋅−2 log
⎛
⎜
⎝
+――
λ
3.7
―――
2.51
⋅Re ‾‾f
⎞
⎟
⎠
((11))
The author discoverd that the two versions of the Colebrook-White equations can be
reconciled within 5 significant figures only if the constants given by equations 9 and 10 are
used in Eq. 2 instead of using Eq. 11.
Julio C. Banks, PE
4. Colebrook-White Friction Factor Equation Derivation 2 of 3.mcdx Page 1 of 2
Reference: "ASHRAE HVAC 2001 Fundamental Handbook" Equation 29b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
D
ε
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎜
⎝
+1 ――――
9.311
⋅⋅Re ―
ε
D
‾‾f
⎞
⎟
⎟
⎠
((29 b))
The constants 1.138 and 9.311 are results of the reconciliation by the author of the
contributing equations. Transform Equation 29b to equation 2
=λ ―
ε
D
((1))
=――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎜
⎝
+⋅2 λ ―――
b
1
⋅Re ‾‾f
⎞
⎟
⎟
⎠
((2))
Symbolic Solution
Substitute Eq. 1 into Eq. 29 b
=――
1
‾f
−
⎛
⎜⎝
+1.138 ⋅2 log
⎛
⎜⎝
―
1
λ
⎞
⎟⎠
⎞
⎟⎠
⋅2 log
⎛
⎜
⎝
+1 ――――
9.311
⋅⋅Re λ ‾‾f
⎞
⎟
⎠
((3))
=――
1
‾f
−(( −1.138 ⋅2 log((λ)))) ⋅2 log
⎛
⎜
⎝
⋅
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
―
1
λ
⎞
⎟
⎠
=――
1
‾f
−1.138 ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((4))
Let =−1.74 ⋅2 log((x)) 1.138 ((5))
Solve for x
≔x =10
⎛
⎜⎝
――
0.602
2
⎞
⎟⎠
2.000 ((6))
Substitute Eq. 5 into Eq. 4
=――
1
‾f
−−1.74 ⋅2 log((x)) ⋅2 log
⎛
⎜
⎝
+λ ―――
9.311
⋅Re ‾‾f
⎞
⎟
⎠
((7))
Julio C. Banks, PE
5. Colebrook-White Friction Factor Equation Derivation 2 of 3.mcdx Page 2 of 2
=――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎝
+⋅x λ ―――
⋅9.311 x
⋅Re ‾f
⎞
⎟
⎠
((8))
Recall Equaton 2
=――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎝
+⋅2 λ ―――
b1
⋅Re ‾‾f
⎞
⎟
⎠
((2))
≔b1 =⋅9.311 x 18.62 ((9))
The standard Colebrook-White equation (used in the generation of the Moody Diagram) is
Colebrook-White Form-1 Equation: =――
1
‾f
⋅−2 log
⎛
⎜
⎜
⎝
+―
λ
a
0
―――
b
0
⋅Re ‾f
⎞
⎟
⎟
⎠
((10))
≡a
0
b
0
⎡
⎣
⎤
⎦
3.707 2.512[[ ]]
An alternate equivalent-form of the Colebrook-White equation 10 is
Colebrook-White Form-2 Equation: =――
1
‾f
−1.74 ⋅2 log
⎛
⎜
⎜
⎝
+⋅2 λ ―――
b
1
⋅Re ‾‾f
⎞
⎟
⎟
⎠
((11))
=b1 18.62
The author discoverd that the two versions of the Colebrook-White equations, form-1 (Eq. 10)
and form-2 (Eq. 11) can be reconciled within 5 significant figures only if the constants given in
this reconciliation report are used instead of the original constant parameters pubished by
Colebrook.
Julio C. Banks, PE
6. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 1 of 5
Derivation of the Colebrook transformation of the Prandtl's equation for the friction factor
in smooth pipes in turbulent flow
by Julio C. Banks, PE
Reference
"Fundamentals of Pipe Flow" by Robert P. Bennedict. ISBN 0-471-03375-8
The Prandtl's equation for the friction factor in smooth pipes in turbulent flow is
=――
1
‾‾fS
−⋅2 log⎛
⎝ ⋅Re ‾‾fS
⎞
⎠ 0.8 ((6.12))
(Page 235 of the reference)
Colebrook took the Prandtl's Equation (6.12) and transformed it into a form
having the constant term, 1.74. Derive Colebrook's transformation of the
Prandtl's equation.
Let
=−0.8 −1.74 ⋅2 log((x)) ((1))
Solve for x from Eq. 1
=⋅2 log((x)) +1.74 0.8
=⋅2 log((x)) 2.54
≔x =10
――
2.54
2
18.62 ((2))
Substitute Eq. 1 into Eq. 6.12
=――
1
‾‾fS
+⋅2 log⎛
⎝ ⋅Re ‾‾fS
⎞
⎠ (( −1.74 ⋅2 log((x))))
=――
1
‾‾fS
−−1.74 ⋅2 log((x)) ⋅2 log
⎛
⎜
⎝
―――
1
⋅Re ‾‾fS
⎞
⎟
⎠
=――
1
‾‾fS
−1.74 ⋅2 log
⎛
⎜
⎝
―――
x
⋅Re ‾‾fS
⎞
⎟
⎠
((3))
Julio C. Banks, PE
7. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 2 of 5
Substitute the numberical value of the constant parameter, x, from Eq. 3 into Eq. 4
=――
1
‾‾fS
−1.74 ⋅2 log
⎛
⎜
⎝
―――
18.62
⋅Re ‾‾fS
⎞
⎟
⎠
((4))
Equation 4 is the Colebrook transformation (first equation on page 239 of the reference) of
the Prandtl's equation 6.12. Subsequently, Colebrook took Eq. 4 and and transformed it into
the form:
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
α
⋅Re ‾‾fS
⎞
⎟
⎠
((5))
Let =⋅−2 log((y)) 1.74 ((6))
Solve for y from Eq. 6
≔y =10
−
⎛
⎜⎝
――
1.74
2
⎞
⎟⎠
0.1349 ((7))
Substitute Eq. 6 into Eq. 4
=――
1
‾‾fS
−⋅−2 log((y)) ⋅2 log
⎛
⎜
⎝
―――
18.62
⋅Re ‾‾fS
⎞
⎟
⎠
((4))
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
⋅18.62 y
⋅Re ‾‾fS
⎞
⎟
⎠
((8))
Comparing equations 5 and 8 it can be seen that
≔α =⋅18.62 y 2.512 ((9))
Substitute Eq. 9 into Eq. 5 we get the following equation
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
2.512
⋅Re ‾‾fS
⎞
⎟
⎠
((10))
Julio C. Banks, PE
8. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 3 of 5
Colbrook took the von Karman's fully rough pipe law of friction, namely Eq. 6.15 (page
257 of the reference) and transformed it into Eq. 11
=――
1
‾‾fR
+⋅2 log
⎛
⎜⎝
―
R
ε
⎞
⎟⎠
1.74 ((6.15))
=――
1
‾‾fR
⋅−2 log
⎛
⎜
⎜
⎝
―
―
ε
D
β
⎞
⎟
⎟
⎠
((11))
Let =λ ―
ε
D
Substitue Eq. 12 into Eq. 6.15 and 11
=――
1
‾‾fR
+⋅−2 log(( ⋅2 λ)) 1.74 ((12))
=――
1
‾‾fR
⋅−2 log
⎛
⎜⎝
―
λ
β
⎞
⎟⎠
((13))
Substitute Eq. 6 into Eq. 12
=――
1
‾‾fR
+⋅−2 log(( ⋅2 λ)) ⋅−2 log((y)) ((14))
=――
1
‾‾fR
⋅−2 log(( ⋅⋅2 y λ)) ((15))
Comparing equations 13 and 15 we obtain the -parameterβ
≔β =――
1
⋅2 y
3.707 ((16))
Substitute Eq. 16 into Eq. 13
=――
1
‾‾fR
⋅−2 log
⎛
⎜⎝
――
λ
3.707
⎞
⎟⎠
((17))
Julio C. Banks, PE
9. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 4 of 5
Colebrook then took equations 5 and 13 and combined them into a single expression
=――
1
‾‾fS
⋅−2 log
⎛
⎜
⎝
―――
α
⋅Re ‾‾fS
⎞
⎟
⎠
((5))
≔α =⋅18.62 y 2.512 ((9))
=――
1
‾‾fR
⋅−2 log
⎛
⎜⎝
―
λ
β
⎞
⎟⎠
((13))
≔β =――
1
⋅2 y
3.707 ((16))
Solve for the arguments of the logarithms for subsequent combination
=10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾‾fR
⎞
⎟
⎠
―
λ
β
((17))
=10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fS
⎞
⎟
⎠
―――
α
⋅Re ‾‾fS
((18))
Add equations 17 and 18
=+10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾‾fR
⎞
⎟
⎠
10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fS
⎞
⎟
⎠
+―
λ
β
―――
α
⋅Re ‾‾fS
((19))
Let =10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fT
⎞
⎟
⎠
+10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fR
⎞
⎟
⎠
10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fS
⎞
⎟
⎠
((20))
Where is the Transition Friction Factor.fT
Substitute Eq. 20 into Eq. 19
=10
⎛
⎜
⎝
――――
1
⋅−2 ‾‾fT
⎞
⎟
⎠
+―
λ
β
―――
α
⋅Re ‾‾fT
((21))
Julio C. Banks, PE
10. Colebrook-White Friction Factor Equation Derivation 3 of 3.mcdx Page 5 of 5
Solve for from Eq. 21――
1
‾‾fT
=――
1
‾‾fT
⋅−2 log
⎛
⎜
⎝
+―
λ
β
―――
α
⋅Re ‾‾fT
⎞
⎟
⎠
((22))
Where
≔α =⋅18.62 y 2.512 ((9))
≔β =――
1
⋅2 y
3.707 ((16))
Julio C. Banks, PE
11. FUNDAMENTALS OF
PIPE FLOW
Julio e. :Bank:S.
Robert P. Benedict
Fellow Mechanical Engineer
Westinghouse Electric Corporation
Steam Turbine Division
Adjunct Professor ofMechanical Engineering
Drexel University
Evening College
Philadelphia, Pennsylvania
A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY & SONS
New York • Chichester • Brisbane • Toronto • Singapore
3
14. Errata
The numerical results do not correspond to
the Colebrook Equation 6.19 (page 240) but
the alternative Form-1 Equation on page 239
which I labeled 6.19b.