Unit 43: Plant and Prcess Principles Unit cde: H/601 44 QCF level: 5 Credit value: 15 OUTCOME 3 - STATIC AND DYNAMIC FLUID SYSTEMS TUTORIAL 3 - VISCOSITY 3 Understand static and namic fluid systems with reference t plant engineering Immersed surfaces: rectangular and circular surfaces, including retaining walls, tank sides, sluice gates, inspectin cvers, valve flanges; hydrstatic pressure and thrust n immersed surfaces Centre f pressure: use f parallel axis therem fr immersed rectangular and circular surfaces Viscsity shear stress; shear rate; namic viscsity: kinematic viscsity Pipeline flw: head lsses e.g. Bernulli s equatin and determinatin f head lss in pipes by D Arcy s frmula; M diagram; head lss due t sudden enlargement and cntractin f pipe diameter; head lss at entrance t a pipe; head lss in valves; Reynlds number; inertia and viscus resistance frces; laminar and turbulent flw; critical velcities Impact f a jet: pwer f a jet nrmal thrust n a mving flat vane; thrust n a mving hemispherical cup; velcity diagrams t determine thrust n mving curved vanes; fluid frictin lsses; system efficiency On cmpletin f this tutrial yu shuld be able t d the fllwing. Define viscsity and its units. Define a Newtnian fluid. Let's start by examining the meaning f viscsity. D.J.DUNN www.freestu.c.uk 1
1. VISCOSITY 1.1 BASIC THEORY Mlecules f fluids exert frces f attractin n each ther. In liquids this is strng enugh t keep the mass tgether but nt strng enugh t keep it rigid. In gases these frces are very weak and cannt hld the mass tgether. When a fluid flws ver a surface, the layer next t the surface may becme attached t it (it wets the surface). The layers f fluid abve the surface are mving s there must be shearing taking place between the layers f the fluid. Fig.1 Let us suppse that the fluid is flwing ver a flat surface in laminated layers frm left t right as shwn in figure 1. y is the distance abve the slid surface (n slip surface) L is an arbitrary distance frm a pint upstream. is the thickness f each layer. is the length f the layer. dx is the distance mved by each layer relative t the ne belw in a crrespnding time dt. u is the velcity f any layer. du is the increase in velcity between tw adjacent layers. Each layer mves a distance dx in time dt relative t the layer belw it. The rati dx/dt must be the change in velcity between layers s du = dx/dt. When any material is defrmed sideways by a (shear) frce acting in the same directin, a shear stress is prduced between the layers and a crrespnding shear strain is prduced. Shear strain is defined as fllws. sidewaysdefrmatin height f the layer being defrmed dx The rate f shear strain is defined as fllws. shear strain time taken dt dx dt du D.J.DUNN www.freestu.c.uk
It is fund that fluids such as water, il and air, behave in such a manner that the shear stress between layers is directly prprtinal t the rate f shear strain. cnstant x Fluids that bey this law are called NEWTONIAN FLUIDS. It is the cnstant in this frmula that we knw as the namic viscsity f the fluid. DYNAMIC VISCOSITY = shear stress rate f shear du FORCE BALANCE and VELOCITY DISTRIBUTION A shear stress exists between each layer and this increases by d ver each layer. The pressure difference between the dwnstream end and the upstream end is. The pressure change is needed t vercme the shear stress. The ttal frce n a layer must be zer s balancing frces n ne layer (assumed 1 m wide) we get the fllwing. d 0 d It is nrmally assumed that the pressure declines unifrmly with distance dwnstream s the pressure gradient is assumed cnstant. The minus sign indicates that the pressure falls with distance. Integrating between the n slip surface (y = 0) and any height y we get du d d d u...(.1) Integrating twice t slve u we get the fllwing. du y A y u Ay B A and B are cnstants f integratin that shuld be slved based n the knwn cnditins (bundary cnditins). Fr the flat surface cnsidered in figure 1 ne bundary cnditin is that u = 0 when y = 0 (the n slip surface). Substitutin reveals the fllwing. 0 = 0 +0 +B hence B = 0 At sme height abve the surface, the velcity will reach the mainstream velcity u. This gives us the secnd bundary cnditin u = u when y =. Substituting we find the fllwing. D.J.DUNN www.freestu.c.uk 3
u A u A hence y u u u y u y Pltting u against y gives figure. BOUNDARY LAYER. The velcity grws frm zer at the surface t a maximum at height. In thery, the value f is infinity but in practice it is taken as the height needed t btain 99% f the mainstream velcity. This layer is called the bundary layer and is the bundary layer thickness. It is a very imprtant cncept and is discussed mre fully in chapter 3. The inverse gradient f the bundary layer is du/ and this is the rate f shear strain. 1.. UNITS f VISCOSITY Fig. 1..1 DYNAMIC VISCOSITY µ The units f namic viscsity µ are N s/m. It is nrmal in the internatinal system (SI) t give a name t a cmpund unit. The ld metric unit was a ne.s/cm and this was called a POISE after Piseuille. It fllws that the SI unit is related t the Pise such that 10 Pise = 1 Ns/m This is nt an acceptable multiple. Since, hwever, 1 CentiPise (1cP) is 0.001 N s/m then the cp is the accepted SI unit. 1cP = 0.001 N s/m. is als cmmnly used fr namic viscsity. There are ther ways f expressing viscsity and this is cvered next. D.J.DUNN www.freestu.c.uk 4
1.. KINEMATIC VISCOSITY namic viscsity This is defined as fllws. density The basic units are m/s. The ld metric unit was the cm/s and this was called the STOKE after the British scientist. It fllws that 1 Stke (St) = 0.0001 m/s and this is nt an acceptable SI multiple. The centistke (cst),hwever, is 0.000001 m/s and this is an acceptable multiple. 1..3 OTHER UNITS 1cSt = 0.000001 m/s = 1 mm/s Other units f viscsity have cme abut because f the way viscsity is measured. Fr example REDWOOD SECONDS cmes frm the name f the Redwd viscmeter. Other units are Engler Degrees, SAE numbers and s n. Cnversin charts and frmulae are available t cnvert them int useable engineering r SI units. 1..4 VISCOMETERS The measurement f viscsity is a large and cmplicated subject. The principles rely n the resistance t flw r the resistance t mtin thrugh a fluid. Many f these are cvered in British Standards 188. The fllwing is a brief descriptin f sme types. U TUBE VISCOMETER The fluid is drawn up int a reservir and allwed t run thrugh a capillary tube t anther reservir in the ther limb f the U tube. The time taken fr the level t fall between the marks is cnverted int cst by multiplying the time by the viscmeter cnstant. = ct The cnstant c shuld be accurately btained by calibrating the viscmeter against a master viscmeter frm a standards labratry. Fig. 3 REDWOOD VISCOMETER This wrks n the principle f allwing the fluid t run thrugh an rifice f very accurate size in an agate blck. 50 ml f fluid are allwed t empty frm the level indicatr int a measuring flask. The time taken is the viscsity in Redwd secnds. There are tw sizes giving Redwd N.1 r N. secnds. These units are cnverted int engineering units with tables. Fig. 4 D.J.DUNN www.freestu.c.uk 5
FALLING SPHERE VISCOMETER Fig. 5 This viscmeter is cvered in BS188 and is based n measuring the time fr a small sphere t fall in a viscus fluid frm ne level t anther. The buyant weight f the sphere is balanced by the fluid resistance and the sphere falls with a cnstant velcity. The thery is based n Stke s Law and is nly valid fr very slw velcities. The thery is cvered later in the sectin n laminar flw where it is shwn that the terminal velcity (u) f the sphere is related t the namic viscsity () and the density f the fluid and sphere ( f and s) by the frmula = F gd (s -f)/18u F is a crrectin factr called the Faxen crrectin factr, which takes int accunt a reductin in the velcity due t the effect f the fluid being cnstrained t flw between the wall f the tube and the sphere. ROTATIONAL TYPES There are many types f viscmeters, which use the principle that it requires a trque t rtate r scillate a disc r cylinder in a fluid. The trque is related t the viscsity. Mdern instruments cnsist f a small electric mtr, which spins a disc r cylinder in the fluid. The trsin f the cnnecting shaft is measured and prcessed int a digital readut f the viscsity in engineering units. Yu shuld nw find ut mre details abut viscmeters by reading BS188, suitable textbks r literature frm il cmpanies. SELF ASSESSMENT EXERCISE N. N. 1 1. Describe the principle f peratin f the fllwing types f viscmeters. a. Redwd Viscmeters. b. British Standard 188 glass U tube viscmeter. c. British Standard 188 Falling Sphere Viscmeter. d. Any frm f Rtatinal Viscmeter D.J.DUNN www.freestu.c.uk 6