ENGINEERING COUNCIL CERTIFICATE LEVEL THERMODYNAMIC, FLUID AND PROCESS ENGINEERING C106 TUTORIAL 5 THE VISCOUS NATURE OF FLUIDS On cmpletin f this tutrial yu shuld be able t d the fllwing. Define viscsity and its units. Define a Newtnian fluid. Describe a range f Viscmeters Let's start by examining the meaning f viscsity.
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 τ is prduced between the layers and a crrespnding shear strain γ is prduced. Shear strain is defined as fllws. γ sideways defrmatin height f the layer being defrmed dx The rate f shear strain is defined as fllws. γ& shear strain time taken γ dt dx dt du
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. τ cnstant x γ& 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 µ shear stress rate f shear τ γ & τ du FORCE BALANCE and VELOCITY DISTRIBUTION A shear stress τ exists between each layer and this increases by dτ 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. + dτ 0 dτ 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 µ dτ d u µ......(.1) Integrating twice t slve u we get the fllwing. du y µ + A y µ 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 δ abve the surface, the velcity will reach the mainstream velcity u. This gives us the secnd bundary cnditin u u when y δ. Substituting we find the fllwing. δ µ u + Aδ δ µ u A hence δ y u y δ µ δ µ u + u + δ µ u δ y Pltting u against y gives figure.. BOUNDARY LAYER. The velcity grws frm zer at the surface t a maximum at height δ. In thery, the value f δ 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 δ 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 γ. 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. The symbl η is als cmmnly used fr namic viscsity. There are ther ways f expressing viscsity and this is cvered next.
1.. KINEMATIC VISCOSITY ν namic viscsity µ This is defined as fllws. ν density ρ 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. 1cSt 0.000001 m/s 1 mm/s 1..3 OTHER UNITS 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 Fig..3 REDWOOD 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. ν ct The cnstant c shuld be accurately btained by calibrating the viscmeter against a master viscmeter frm a standards labratry. 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
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 (µ) and the density f the fluid and sphere (ρ f and ρs) by the frmula µ F gd (ρs -ρf)/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..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