UNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of bonary layer Thickness an classification Displacement an momentm Thickness Development of laminar an trblent flows in circlar pipes Major an minor Losses of flow in pipes Pipes in series an in parallel Pipe network Bonary Layers Fli flowing over a stationary srface, e.g. the be of a river, or the wall of a pipe, is broght to rest by the shear stress to,this gives a, now familiar, velocity profile: Zero at the wall A maximm at the centre of the flow.the profile oesn t jst exit. It is bil p graally. Starting when it first flows past the srface e.g. when it enters a pipe. Consiering a flat plate in a fli. Upstream the velocity profile is niform,this is known as free stream flow. Downstream a velocity profile exists.this is known as flly evelope flow. BOUNDARY LAYER FLOW: When a real fli flow past a soli boy or a soli wall, the fli particles ahere to the bonary an conition of no slip occrs. This means that the velocity of fli close to the bonary will be same as that of the bonary. If the bonary is stationary, the velocity of fli at the bonary will be zero. Farther away from the bonary, the velocity will be higher an as a reslt of this variation of velocity, the velocity graient will exit. The velocity of fli increases from zero velocity on the stationary y

bonary to free stream velocity (U) of the fli in the irection normal to the bonary. This variation of velocity from zero to free stream velocity in the irection normal to the bonary takes place in a narrow region in the vicinity of soli bonary. This narrow region of the fli is calle bonary layer. The theory ealing with bonary layer flows is calle bonary layer theory. Accoring to bonary layer theory, the flow of fli in the neighborhoo of the soli bonary may be ivie into two regions as shown in figre.. A very thin layer of the fli, calle the bonary layer, in the immeiate neighborhoo of the soli bonary, where the variation of velocity from zero at the soli bonary to free stream velocity in the irection normal to the bonary takes place. In this region, the velocity graient exists an hence the fli exerts a shear stress on the wall in the irection of motion. The t vale of shear stress is given by y. The remaining fli, which is otsie the bonary layer. The velocity otsie the bonary layer is constant an eqal to free-stream velocity. As there is no variation of velocity in this region, the velocity graient becomes zero. As a reslt of this shear stress is zero. t Laminar Bonary Layer: For efining the bonary layer (i.e., laminar bonary layer or trblent bonary layer) consier the flow of a fli, having a free-stream velocity (U), over a smooth thin plate which is flat an parallel to the irection for free stream of fli as shown in figre. Let s consier the flow with zero pressre graient on one sie of the plate, which is stationary. The velocity of fli on the srface of the plate shol be eqal to the velocity of the plate. Bt plate is stationary an hence velocity of fli on the srface of the plate is zero. Bt at a istance away from the plate, the fli is having certain velocity. Ths a velocity graient is set p in the fli near the srface of the plate. This velocity graient evelops shear resistance, which retars the fli. Ths the fli with a niform free stream velocity (U) is retare in the vicinity of the soli srface of the plate an the bonary layer region begins at the sharp leaing ege. At sbseqent points ownstream the leaing ege, the bonary layer region increases becase the retare fli is frther retare. This is also referre as the growth of bonary layer. Near the leaing ege of the srface of the plate, where the thickness is small, the flow in the bonary layer is laminar thogh the main flow is trblent. This layer of fli is sai to be laminar bonary layer. This is shown by AE in figre. The length of the plate from the leaing ege, pto which laminar bonary layer exists, is calle laminar zone. This is shown by istance AB. The istance of B from leaing ege is obtaine from Reynol s nmber eqal to 5 x 0 5 for a plate. Becase pto this Reynol s nmber the bonary layer is laminar. The Reynol s nmber is given by R e x U x Where x= Distance from leaing ege U= Free-stream velocity of fli

= Kinematic viscosity of fli Hence, for laminar bonary layer, we have 5 U x 50 ------------- () If the vales of U an are known, x or the istance from the leaing ege pto which laminar bonary layer exists can be calclate Trblent Bonary Layer: If the length of the plate is more than the istance x, calclate from eqation (), the thickness of bonary layer will go on increasing in the own-stream irection. Then the laminar bonary layer becomes nstable an motion of fli within it, is istrbe an irreglar which leas to a transition from laminar to trblent bonary layer. This short length over which the bonary layer flow changes from laminar to trblent is calle transition zone. This is shown by istance BC in figre. Frther ownstream the transition zone, the bonary layer is trblent an continos to grow in thickness. This layer of bonary is calle trblent bonary layer, which is shown by the portion FG in figre.. Laminar Sb-layer: This is the region in the trblent bonary layer bonary layer zone, ajacent to the soli srface of the plate as shown in figre. In this zone, the velocity variation is inflence only by viscos effects. Thogh the velocity istribtion wol be a parabolic crve in the laminar sb-layer zone, bt in view of the very small thickness we can reasonably assme that velocity variation is linear an so the velocity graient can be consiere constant. Therefore, the shear stress in the laminar sb-layer wol be constant an eqal to the bonary shear stress. 0 Ths the shear stress in the sb-layer is y 0 y0 y { For linear variation, y y } Bonary Layer Thickness(): It is efine as the istance from bonary of the soli boy measre in y-irection to the point, where the velocity of the fli is approximately eqal to 0.99 times the free stream (U) velocity of the fli. It is enote by the symbol. For laminar an trblent zone it is enote as:. lam= Thickness of laminar bonary layer,. tr=thickness of trblent bonary layer, an 3. = Thickness of laminar sb-layer. Displacement thickness ( *):It is efine as the istance, perpeniclar to the bonary of the soli boy, by which the bonary shol be isplace to compensate for the rection in flow rate on accont of bonary layer formation. It is enote by the symbol *. It is also efine as: The istance, perpeniclar to the bonary, by which the free stream is isplace e to the formation of bonary layer. * y 0 U

Momentm thickness (θ):it is efine as the istance, measre perpeniclar to the bonary of the soli boy, by which the bonary shol be isplace to compensate for the rection in momentm of the flowing fli on accont of bonary layer formation. It is enote by the symbol θ. 0 U U y Energy thickness ( **):It is efine as the istance, perpeniclar to the bonary of the soli boy, by which the bonary shol be isplace to compensate for the rection in kinetic energy of the flowing fli on accont of bonary layer formation. It is enote by the symbol **. ** U U 0 y Bonary layers in pipes Initially of the laminar form. It changes epening on the ratio of inertial anviscos forces; i.e. whether we have laminar (viscos forces high) or trblent flow (inertial forces high). Use Reynols nmber to etermine which state. Laminar flow: profile parabolic (prove in earlier lectres) The first part of the bonary layer growth iagram. Trblent (or transitional), Laminar an the trblent (transitional) zones of the bonary layer growth iagram. Length of pipe for flly evelope flow is the entry length. Laminar flow =0 * iameter

Trblent flow = 60 * iameter Bonary layer separation Divergent flows: Positive pressre graients.pressre increases in the irection of flow. The fli in the bonary layer has so littlemomentm that it is broght to rest,an possibly reverse in irection.reversal lifts the bonary layer. Expression for loss of hea e to Friction in pipes (Darcy weisbach s Eqation): h f 4.f g LV 4f.L.V g The above eqation is known as Darcy- weisbach s eqation. This is commonly se for fining loss of hea e to friction in pipes. Eqation (5) is written as h f f.l.v g Then f is known as friction factor. HYDRAULIC GRADIENT AND TOTAL ENERGY LINE: This concept of hyralic graient line an total energy line is very sefl in the sty of flow of flis throgh pipes. They are efine as.hyralic Graient Line: It is efine as the line which gives the sm of pressre hea (p/w) an atm hea (z) of a flowing fli in a pipe with respect to some reference line or it is the line which is obtaine by joining the top of all vertical orinates, showing the pressre hea (p/w) of a flowing fli in a pipe from the centre of the pipe. It is briefly written as H.G.L (Hyralic Graient Line)..Total Energy Line: It is efine as the line which gives the sm of pressre hea, atm hea an kinetic hea of a flowing fli in a pipe with respect to some reference line. It is also efine as the line

which is obtaine by joining the tops of all vertical orinates showing the sm of pressre hea an kinetic hea from the centre of the pipe. It is briefly written as T.E.L (Total Energy Line) DARCY WEISBACH EQUATION (Derivation refer class notes) Mooy iagram for friction factor: Minor losses: (Derivation an formlas refer class notes) Sen enlargement Sen contraction Sen obstrction Entrance in pipe Exit in pipe Losses by ben Losses by sing fittings FLOW THROUGH PIPES IN SERIES OR FLOW THROUGH COMPOUND PIPES: H 4fL V g 4fL V + g 4fL3V3 + g 3 = 4f g LV FLOW THROUGH PARALLEL PIPES: L V L3V 3 3 Loss of hea for branch pipe = Loss of hea for branch pipe

or 4f L V g = 4f L V g If f=f, then LV = g L V g UNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of bonary layer Thickness an classification Displacement an momentm Thickness Development of laminar an trblent flows in circlar pipes Major an minor Losses of flow in pipes Pipes in series an in parallel Pipe network Bonary layer concepts (Theory) (A Text book of fli mechanics by R.K.Rajpt Page no: 76 an 77) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 6 to 63) Bonary layer thickness. Bonary layer thickness (Derivation Refer Class Notes) (A Text book of fli mechanics by R.K.Rajpt Page no: 70 to79, Example: 3. to 3.9) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 66 to 68, Problem: 3. an 3.) Drag force on a flat plate by momentm eqation (A Text book of fli mechanics by R.K.Rajpt Page no: 79 to76, Example: 3.0 to 3.8) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 69 to 648, Problem: 3.3 an 3.7) Hyralic an energy graient (A Text book of fli mechanics by R.K.Rajpt Page no: 648 to 658, Example:.6 to.) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 49 to 498, Problem:. to.6) Darcy Weisbach eqation an chezy s eqn. (Derivation Refer Class Notes) (A Text book of fli mechanics by R.K.Rajpt Page no: 630 to 633, Example:. to.4) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 467 to 47, Problem:. to.7) Friction factor an Mooy iagram. (Notes) Minor losses (Derivation refer class notes) (A Text book of fli mechanics by R.K.Rajpt Page no: 636 to 648, Example:.8 to.5) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 47 to 49, Problem:.8 to.) Flow thogh pipes in series (A Text book of fli mechanics by R.K.Rajpt Page no: 659 to 665, Example:. to.6)

(A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 50 to 508, Problem:.30 to.3) Flow thogh pipes in parallel. (A Text book of fli mechanics by R.K.Rajpt Page no: 665 to 684, Example:.7 to.39) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 508 to 54, Problem:.3 to.4) Flow thogh branche pipes. (A Text book of fli mechanics by R.K.Rajpt Page no: 684 to 687, Example:.40 to.4) (A Text book of fli mechanics an hyralic machines by Dr.R.K.Bansal Page no: 54 to 530, Problem:.4 to.44) TWO MARKS BOUNDARY LAYER & FLOW THROUGH PIPES. Mention the range of Reynol s nmber for laminar an trblent flow in a pipe. If the Reynols nmber is less than 000, the flow is laminar. Bt if the Reynol s nmber is greater than 4000, the flow is trblent flow.. What oes Haigen-Poiselle eqation refer to? The eqation refers to the vale of loss of hea in a pipe of length L e to viscosity in a laminar flow. 3. What is Hagen poiseille s formla? (P -P ) / ρg = h f = 3 μūl / ρgd The expression is known as Hagen poiseille formla. Where P -P / ρg = Loss of pressre hea, Ū= Average velocity, μ = Coefficient of viscosity, D = Diameter of pipe, L = Length of pipe 4. Write the expression for shear stress? Shear stress ζ = - ( p/ x) (r/) ζmax = - ( p/ x) (R/) 5. Give the formla for velocity istribtion: - The formla for velocity istribtion is given as = - (¼ μ) ( p/ x) (R -r ) Where R = Rais of the pipe, r = Rais of the fli element 6. Give the eqation for average velocity: - The eqation for average velocity is given as Ū = - (/8μ) ( p/ x) R Where R = Rais of the pipe 7. Write the relation between Umax an Ū? Umax / Ū = { - (¼ μ) ( p/ x) R } / { - ⅛μ ( p/ x) R } Umax / Ū = 8. Give the expression for the coefficient of friction in viscos flow? Coefficient of friction between pipe an fli in viscos flow f =6/ Re Where, f = Re = Reynols nmber 9. What are the factors to be etermine when viscos fli flows throgh the circlar pipe? The factors to be etermine are: i. Velocity istribtion across the section. ii. Ratio of maximm velocity to the average velocity.

iii. Shear stress istribtion. iv. Drop of pressre for a given length 0. Define kinetic energy correction factor? Kinetic energy factor is efine as the ratio of the kinetic energy of the flow per sec base on actal velocity across a section to the kinetic energy of the flow per sec base on average velocity across the same section. It is enote by (α). K. E factor (α) = K.E per sec base on actal velocity / K.E per sec base on Average velocity. Define momentm correction factor (β): It is efine as the ratio of momentm of the flow per sec base on actal velocity to the momentm of the flow per sec base on average velocity across the section. β= Momentm per sec base on actal velocity/momentm Per sec base on average velocity. Define Bonary layer. When a real fli flow passe a soli bonary, fli layer is ahere to the soli bonary. De to ahesion fli nergoes retaration thereby eveloping a small region in the immeiate vicinity of the bonary. This region is known as bonary layer. 3. What is mean by bonary layer growth? At sbseqent points ownstream of the leaing ege, the bonary layer region increases becase the retare fli is frther retare. This is referre as growth of bonary layer. 4. Classification of bonary layer. (i) Laminar bonary layer, (ii) Transition zone, (iii) Trblent bonary layer. 5. Define Laminar bonary layer. Near the leaing ege of the srface of the plate the thickness of bonary layer is small an flow is laminar. This layer of fli is sai to be laminar bonary layer. The length of the plate from the leaing ege, pto which laminar bonary layer exists is calle as laminar zone. In this zone the velocity profile is parabolic. 6. Define transition zone. After laminar zone, the laminar bonary layer becomes nstable an the fli motion transforme to trblent bonary layer. This short length over which the changes taking place is calle as transition zone. 7. Define Trblent bonary. Frther ownstream of transition zone, the bonary layer is trblent an continos to grow in thickness. This layer of bonary is calle trblent bonary layer. 8. Define Laminar sb Layer In the trblent bonary layer zone, ajacent to the soli srface of the plate the velocity variation is inflence by viscos effects. De to very small thickness, the velocity istribtion is almost linear. This region is known as laminar sb layer. 9. Define Bonary layer Thickness. It is efine as the istance from the soli bonary measre in y-irection to the point, where the velocity of fli is approximately eqal to 0.99 times the free stream velocity (U) of the fli. It is enote by δ. 0. List the varios types of bonary layer thickness. Displacement thickness(δ*), Momentm thickness(θ), Energy thickness(δ**). Define isplacement thickness. The isplacement thickness (δ) is efine as the istance by which the bonary shol be isplace to compensate for the rection in flow rate on accont of bonary layer formation. δ* = [ (/U) ] y. Define momentm thickness.

The momentm thickness (θ) is efine as the istance by which the bonary shol be isplace to compensate for the rection in momentm of the flowing fli on accont of bonary layer formation. θ = [ (/U) (/U) ] y 3. Define energy thickness The energy thickness (δ**) is efine as the istance by which the bonary shol be isplace to compensate for the rection in kinetic energy of the flowing fli on accont of bonary layer formation. δ** = [ (/U) (/U) 3 ] y 4. What is meant by energy loss in a pipe? When the fli flows throgh a pipe, it looses some energy or hea e to frictional resistance an other reasons. It is calle energy loss. The losses are classifie as; Major losses an Minor losses 5. Explain the major losses in a pipe. The major energy losses in a pipe is mainly e to the frictional resistance case by the shear force between the fli particles an bonary walls of the pipe an also e to viscosity of the fli. 6. Explain minor losses in a pipe. The loss of energy or hea e to change of velocity of the flowing fli in magnite or irection is calle minor losses. It incles: sen expansion of the pipe, sen contraction of the pipe, ben in a pipe, pipe fittings an obstrction in the pipe, etc. 7. State Darcy-Weisbach eqation OR What is the expression for hea loss e to friction? h f = 4flv / g where, h f = Hea loss e to friction (m), L = Length of the pipe (m), = Diameter of the pipe (m), V = Velocity of flow (m/sec) f = Coefficient of friction 8. What are the factors inflencing the frictional loss in pipe flow? Frictional resistance for the trblent flow is, i. Proportional to v n where v varies from.5 to.0. ii. Proportional to the ensity of fli. iii. Proportional to the area of srface in contact. iv. Inepenent of pressre. v. Depen on the natre of the srface in contact. 9. Write the expression for loss of hea e to sen enlargement of the pipe. h exp = (V -V ) /g Where, h exp = Loss of hea e to sen enlargement of pipe. V = Velocity of flow at pipe ; V = Velocity of flow at pipe. 30. Write the expression for loss of hea e to sen contraction. h con =0.5 V /g h con = Loss of hea e to sen contraction. V = Velocity at otlet of pipe. 3. Write the expression for loss of hea at the entrance of the pipe. hi =0.5V /g hi = Loss of hea at entrance of pipe. V = Velocity of liqi at inlet of the pipe. 3. Write the expression for loss of hea at exit of the pipe. ho = V /g where, ho = Loss of hea at exit of the pipe. V = Velocity of liqi at inlet an otlet of the pipe.

33. Give an expression for loss of hea e to an obstrction in pipe Loss of hea e to an obstrction = V / g ( A/ Cc (A-a ) - ) Where, A = area of pipe, a = Max area of obstrction, V = Velocity of liqi in pipe A-a = Area of flow of liqi at section - 34. What is compon pipe or pipes in series? When the pipes of ifferent length an ifferent iameters are connecte en to en, then the pipes are calle as compon pipes or pipes in series. 35. What is mean by parallel pipe an write the governing eqations. When the pipe ivies into two or more branches an again join together ownstream to form a single pipe then it is calle as pipes in parallel. The governing eqations are: Q = Q + Q 3 h f = h f 36. Define eqivalent pipe an write the eqation to obtain eqivalent pipe iameter. The single pipe replacing the compon pipe with same iameter withot change in ischarge an hea loss is known as eqivalent pipe. L = L + L + L 3 (L/ 5 ) = (L / 5 ) + (L / 5 ) + (L3 / 3 5 ) 37. What is meant by Mooy s chart an what are the ses of Mooy s chart? The basic chart plotte against Darcy-Weisbach friction factor against Reynol s Nmber (Re) for the variety of relative roghness an flow regimes. The relative roghness is the ratio of the mean height of roghness of the pipe an its iameter (ε/d). Mooy s iagram is accrate to abot 5% for esign calclations an se for a large nmber of applications. It can be se for non-circlar conits an also for open channels. 38. Define the terms a) Hyralic graient line [HGL] b) Total Energy line [TEL] Hyralic graient line: It is efine as the line which gives the sm of pressre hea an atm hea of a flowing fli in a pipe with respect the reference line. HGL = Sm of Pressre Hea an Datm hea Total energy line: Total energy line is efine as the line which gives the sm of pressre hea, atm hea an kinetic hea of a flowing fli in a pipe with respect to some reference line. TEL = Sm of Pressre Hea, Datm hea an Velocity hea