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#### partial differential equation

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Book Chapter

By Manas Shirgaokar
Series: ASM Technical Books
Publisher: ASM International
Published: 01 February 2005
DOI: 10.31399/asm.tb.chffa.t51040091
EISBN: 978-1-62708-300-3
... operations. As a result, the effect of temperatures on flow stress and metal flow during hot forming are often not considered adequately. Finite Difference Method In the finite difference method, the derivatives in the governing partial differential equations are written in terms of difference...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 January 2022
DOI: 10.31399/asm.tb.isceg.t59320049
EISBN: 978-1-62708-332-4
... of safety 5.4.4.1 Stress Analysis Finite Difference Method The finite difference method is a direct method of discretizing (assigning specific data values) using partial differential equations ( Ref 3 ). These equations are known as finite-difference equations. The method uses a simplified...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320259
EISBN: 978-1-62708-357-7
... ) are equal. [Answer] Partially differentiating Eq 9.31 and 9.32 with respect to λ, and finding out the extreme value conditions, then the following relations can be obtained. (Eq 9.33)  Maximum growth rate:  ( ∂ v / ∂ λ ) = 0 → Δ G min = 4 σ α / β V / λ...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320167
EISBN: 978-1-62708-357-7
... π ∫ 0 ζ exp ⁡ ( − ζ 2 ) d ζ [Answer] Equation 6.17 can be obtained in various ways, and the conversion of Fick’s second law into an ordinary differential equation using a non-dimensional variable: ζ = z / 2 D t is explained here...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.ssde.t52310011
EISBN: 978-1-62708-286-0
...(a) would be recorded as Fe /O2 /OH ,Fe2+ /Fe . This cell is an example of a differential aeration corrosion cell, which is discussed later. Processes at a single electrode The Nernst Equation Electrochemical reactions require a transfer of charge; hence, there is a coupling between chemical and electrical...
Book Chapter

By Ruediger Rosenkranz
Series: ASM Technical Books
Publisher: ASM International
Published: 01 November 2019
EISBN: 978-1-62708-247-1
... and of the secondary electrons leaving it. The solution of the corresponding differential equation delivers the voltage U c which is building up at the structure. Fig. 7 Equivalent circuit diagram, solution of the corresponding differential equation and graphical representation for different structure...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.ssde.t52310057
EISBN: 978-1-62708-286-0
... υ X V m = N C r 0 3 2 X V m Differentiating Eq 31 with time, equating to Eq 3 , and separating variables results in: (Eq 32) X d X = N O s D O υ N B O d t = N O s D O...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2018
DOI: 10.31399/asm.tb.aceg.t68410151
EISBN: 978-1-62708-280-8
... mechanical properties benefit from these types of inserts. Figure 7.6 shows a concept of a tilt-poured differential carrier fed through a ring feeder on the top flange. The pinion end, which has a mass concentration, is cooled intensely to create partial solidification during pouring and the initial...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 30 September 2023
DOI: 10.31399/asm.tb.stmflw.t59390064
EISBN: 978-1-62708-459-8
..., and the role of elastic deflection and ultrasonic vibration. adsorption boundary lubrication elastohydrodynamic lubrication frictional instabilities hydrodynamic lubrication lubricant entrainment lubrication regimes mixed lubrication plastohydrodynamic lubrication Reynolds equation Stribeck...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 August 2005
DOI: 10.31399/asm.tb.pb.t51230105
EISBN: 978-1-62708-351-5
... All chemical reactions are reversible, including oxidation reactions. In general, the oxidation of any metal can be described by an equation of the form: (Eq 3.1) n M + [ m / 2 ] O 2 ↔ M n O m The reaction will proceed spontaneously in either direction...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.tm.t52320055
EISBN: 978-1-62708-357-7
... components increase by a factor φ, then so does the value of V , the following equation can be obtained: (Eq 3.34) φ ⋅ V ( m A , m B ) = V ( φ m A , φ m B ) Differentiate both sides of Eq 3.34 by φ and then replace φ = 1 to obtain (Eq 3.35...
Book Chapter

By Hugh Reynolds
Series: ASM Technical Books
Publisher: ASM International
Published: 01 September 2011
DOI: 10.31399/asm.tb.cfw.t52860115
EISBN: 978-1-62708-338-6
.... The local α (previously defined) can be better defined taking into account friction on the dome (a much more complicated numerical solution—three nonlinear partial differential equations). This equation is solved down to the inflection point (2/ a = tan 2 (α)) when the local α equals 54.74...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2015
DOI: 10.31399/asm.tb.cpi2.t55030349
EISBN: 978-1-62708-282-2
... corrosivity. The charge-transfer controlled reaction involving carbonic acid and carbon steel (or iron) can be represented in terms of the concentration or partial pressure of dissolved CO 2 in the medium to arrive at a corrosion rate equation that incorporates the order of the reaction and an exponential...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 March 2006
DOI: 10.31399/asm.tb.fdsm.t69870045
EISBN: 978-1-62708-344-7
..., and important refinements and modifications. It demonstrates the use of approximate equations, including the method of universal slopes and the four-point correlation technique, which provides reasonable estimates of elastic and plastic lines from information obtained in standard tensile tests. It also...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.tb.ssde.t52310027
EISBN: 978-1-62708-286-0
... or a mixture of hydrofluoric and nitric acids. Environmental Variables Influencing Uniform Corrosion The corrosion of stainless steels is usually the result of contact with an electrolyte, allowing a complex set of partial electrochemical reactions, which may occur sequentially or concurrently...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 April 2004
DOI: 10.31399/asm.tb.ps.t62440001
EISBN: 978-1-62708-352-2
... for a brief historical background of solders and brazes, respectively). The type of metallurgical reaction between a filler and parent metal is sometimes used to differentiate soldering from brazing. Solders usually react to form intermetallic phases, that is, compounds of the constituent elements...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 August 2005
DOI: 10.31399/asm.tb.pb.t51230001
EISBN: 978-1-62708-351-5
... is sometimes used to differentiate brazing from soldering. Brazes mostly alloy with the parent materials to form solid solutions, which are mixtures of the constituents on an atomic scale. By contrast, solders usually react to form intermetallic phases, that is, compounds of the constituent elements that have...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 January 2015
DOI: 10.31399/asm.tb.spsp2.t54410063
EISBN: 978-1-62708-265-5
... through laths or plates. Fig. 5.4 Surface tilting and its relationship to martensitic structure in an Fe-0.2C alloy. (a) Surface tilting after quenching. (b) Partially polished surface. (c) Area in (b) after etching. (d) Same area after polishing to remove all relief and re-etching. Nital etch...
Book Chapter

Series: ASM Technical Books
Publisher: ASM International
Published: 01 January 2015
DOI: 10.31399/asm.tb.spsp2.t54410099
EISBN: 978-1-62708-265-5
... and incomplete bainite transformation. Source: Ref 6.6 , 6.7 Bainite Transformation Start Temperatures The temperature at which bainite transformation starts is referred to as the B S temperature, and several empirical equations that show the effect of alloying elements on B S have been determined...
Book Chapter

By H. M. Ledbetter
Series: ASM Technical Books
Publisher: ASM International
Published: 01 June 1983
DOI: 10.31399/asm.tb.mlt.t62860001
EISBN: 978-1-62708-348-5
... in a material’s properties. These differential equations also provide useful relationships for computing propagations of known errors. For example, B and ν determined by either resonance or static stress-strain are highly inaccurate compared with pulse-method results. In the pulse case, 1% measurement...