Substantive derivative

In the special case when the velocity components are those of the fluid, the total rate of change of 4> is denoted by D /Dr, which is known as the substantive derivative ... 

Consider the fluid s x-component of motion in a rectangular Cartesian coordinate system. By following the flow, the rate of change of a fluid element s momentum is given by the substantive derivative of the momentum. By Newton s second law of motion, this can be equated to the net force acting on the element. For an element of fluid having volume Sx ySz, the equation of motion can be written for the x-component as follows ... 

In the previous equation, the total derivative is also called Lagrangian derivative, material derivative, substantive derivative, or comoving derivative. The combination of the partial derivative and the convective derivative is also called the Eulerian derivative. Again, it is very important that this equation be thoroughly understood. It is to be noted that in the enviromnental engineering literature, many authors confuse the difference between the total derivative and the partial derivative. Some authors use the partial derivative instead of the total derivative and vise versa. As shown by the previous equation, there is a big difference between the total derivative and the partial derivative. If this difference is not carefully observed, any equation written that uses one derivative instead of the other is conceptually wrong this... 

Brownian diffusion is neglected compared with turbulent transport. The left-hand side represents Dpc/Dt, the Stokes or substantive derivative of p--. The first term on the right-hand side is the turbulent diffusion of The second term —2v p j Vp7 is generally positive and represents the generation of p by transfer from the mean How. The third term. 2p//fl, is the contribution of variations in the mte of gas-to-particle conversion by chemical reaction to the rate of production of p . The la.st term is the decrease of mean square fluctuations pj due to the action of small scale diffusion (dissipation). Thus three types of terms appear on the right-hand side of (13,16), the balance equation for Pi (i) turbulent diffusion of p, and tnmsfer from the mean (low to p.. which alTeci... 

But the left side of Equation 4-104 is just the substantive derivative, also known as the convective or material derivative following a particle. It is usually denoted by the df/dt operator, that is,... 

Direct dyes are defined as anionic dyes substantive to ceUulosic fibers (cotton, viscose, etc), when applied from an aqueous bath containing an electrolyte. Before the discovery of Congo Red in 1884, only mordanted cotton could be dyed. Congo Red [573-58-0] (62) (Cl Direct Red 28 Cl 22120) a primary symmetrical disazo dye, which is made readily from bisdiazotized benzidine and naphthionic acid [84-86-6] (4-arnino-l-naphthalenesulfonic acid), was the precursor of a most important line of dyes, including all shades, derived from benzidine and its homologues. Today, no benzidine dye is produced because benzidine is carcinogenic. 

Many other products can be used as softeners but are less important commercially because of greater cost and/or inferior properties. Examples are anionic surfactants such as long-chain (C16-C22) alkyl sulphates, sulphonates, sulphosuccinates and soaps. These have rather low substantivity and are easily washed out. Nonionic types of limited substantivity and durability, usually applied by padding, include polyethoxylated derivatives of long-chain alcohols, acids, glycerides, oils and waxes. They are useful where ionic surfactants would pose compatibility problems and they exhibit useful antistatic properties, but they are more frequently used as lubricants in combination with other softeners, particularly the cationics. 

Substituted triazinyl derivatives of DAS are usually chosen for pad-dry-bake application to cotton in conjunction with an easy-care or durable-press finish. In these mildly acidic conditions (pH about 4) the FBA must show appreciable resistance towards the catalyst (usually magnesium chloride) necessary to cure the resin. The less substantive products in the upper half of Table 11.1 are important in this respect, as are compounds of type 11.9 where R = OCH3 or CH3NCH2CH2OH. It is likely that the hydroxyethylamino groups present in many of these compounds participate in condensation reactions with N-methylol groups in the cellulose-reactant resin. The performance of an FBA applied in conjunction with a resin finish can be modified and improved by careful formulation of the pad liquor but this lies beyond the scope of the present chapter. Alternatively, FBA and resin can be applied in two separate steps most DAST-type brighteners would be suitable if applied in this way. 

The manufacture of several important brighteners containing alkoxytriazine groups, such as the DAS derivative of highest substantivity in Table 11.1, does not follow the conventional sequence. The first step involves reaction of cyanuric chloride with excess methanol and excess acid acceptor, usually sodium bicarbonate. Under acidic conditions this reaction takes a quite different course and can become dangerously violent (Scheme 11.4). 

A detailed comparison [181] of three vinylsulphone dyes included a low-substantivity monoazo N-acetyl H acid derivative (Cl Reactive Red 35), a monoazo N-acetyl J acid type of higher substantivity (Cl Reactive Orange 82) and a phthalocyanine turquoise somewhat prone to aggregation (Cl Reactive Blue 21). Dyeings of these individual products were subjected to three wash-off procedures ... 

Approximately 75% of fluorescent brighteners belong to the stilbene class. These are almost invariably derived from 4,4, diaminostilbene 2,2, disulphonic acid (1.6 X = NH2), often condensed with cyanuric chloride to take advantage of the further contribution of the s-triazine rings to substantivity for cellulose. 

Since their commercial introduction during the 1940s as components of proprietary detergents and laundry preparations, these products have found extensive usage in the whitening of paper and textile materials. Disperse FBAs are available for whitening hydrophobic fibres and solvent-soluble FBAs impart fluorescence to oils, paints, varnishes and waxes. Approximately 75% of commercially established FBAs are stilbene derivatives with inherent substantivity for paper and cellulosic textiles, but the remainder come from about twenty different chemical classes. These include aminocoumarins (6%), naphthalimides (3%), pyrazoles and pyrazolines (each about 2%), acenaphthenes, benzidine sulphones, stilbene-naphthotriazoles, thiazoles and xanthenes (each about 1%). FBAs of these and other chemical types are discussed in detail in Chapter 11 of Volume 2. 

Table 3.6 Substantivity and structure of p-substituted monoazo naphthionic acid derivatives [68]...

Table 3.8 Substantivity of symmetrical disazo dyes derived from naphthylamine monosulphonic acids [70]...

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