Hagen

The esters obtained by acid or alkaline esterification are, in general, distinguished from the phenolic ethers 

It has been shown that the degree of electrol) ic conductivity of the sodium salt is a certain indication of the basicity of the corresponding acid. The method is of very general application, since insoluble acids 

German silver, platinoid, or manganin, the last is the best on account of its low temperature coefficient. The wire must be calibrated. For high resistances (itooo-200,000 Ohms) the special form of resistance 

The apparatus is arranged in the form of Kirchhoff s modification of the Wlieatstone bridge (Fig. 8), the connections being made with stout copper wire. The induction coil is enclosed in a sound tight case, or is 

The value of k is determined by measuring the conductivity of N/50 solution of potassium chloride, for which Kohlrausch found the values  

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This equation may be integiated and the constant of integration evaluated using the boundary conditions du/and u[R) =0. The solution is the weU-known Hagen-Poiseuihe relationship given by... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

Capillary Viscometers. Capillary flow measurement is a popular method for measuring viscosity (21,145,146) it is also the oldest. A Hquid drains or is forced through a fine-bore tube, and the viscosity is determined from the measured flow, appHed pressure, and tube dimensions. The basic equation is the Hagen-Poiseuike expression (eq. 17), where Tj is the viscosity, r the radius of the capillary, /S.p the pressure drop through the capillary, IV the volume of hquid that flows in time /, and U the length of the capillary. 

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation... 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Simplified theoiy for both batch and continuous filtration is based on the time-honored Hagen-Poiseuille equation ... 

HAGEN, H, Glasfaserverstdrkte Kunstoffe, Springer-Verlag, Berlin (1961)... 

Hagen,. M., "Clathrate Inclusion Compounds", Reinhold, New York, 1962... 

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