* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download First year fluid mechanics
Lattice Boltzmann methods wikipedia , lookup
Coandă effect wikipedia , lookup
Boundary layer wikipedia , lookup
Lift (force) wikipedia , lookup
Airy wave theory wikipedia , lookup
Water metering wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
Hydraulic jumps in rectangular channels wikipedia , lookup
Wind-turbine aerodynamics wikipedia , lookup
Hydraulic machinery wikipedia , lookup
Flow measurement wikipedia , lookup
Navier–Stokes equations wikipedia , lookup
Derivation of the Navier–Stokes equations wikipedia , lookup
Compressible flow wikipedia , lookup
Aerodynamics wikipedia , lookup
Computational fluid dynamics wikipedia , lookup
Flow conditioning wikipedia , lookup
Bernoulli's principle wikipedia , lookup
First year fluid mechanics Flows in pipes and pipelines The steady flow energy equation Bernoulli’s equation is an energy equation derived for frictionless (inviscid) conditions with no energy input or extraction. It is a special form of more general steady flow energy equation, which includes viscose losses and work transfer to the fluid. These effects are accounted for by introducing additional terms into Bernoulli’s equation. Let us consider the balance of energy inside volume abcd (figure 1), which is bounded by the walls of a stream tube and two cross sections 1 and 2 at heights z1,2 from an arbitrary datum level. In this derivation we assume that the fluid velocity does not change across the stream tube. The fluid velocities at the inlet 1 and outlet 2 are V1,2 and the areas of the cross sections are A1,2 . External work is applied to the fluid inside the control volume with power P (e.g. pumps or turbines operate there), and Pf is the power of frictional forces inside the control volume and on its boundaries. During the small time period ∆t the fluid inside the volume a′ add′ enters the control volume and the fluid inside the volume bcc′ b′ leaves the control volume. According the the continuity principles for a steady flow these volumes are the same: A1 V1 ∆t = A2 V2 ∆t = Q ∆t, where Q is the volumetric flow rate of fluid. The energy entering the control volume with the fluid through the cross section 1 is V12 + g z1 ) ρ Q ∆t, 2 and includes the kinetic energy of moving fluid and the potential energy of fluid in the gravitational field. Analogously, the energy leaving the control volume through the cross section 2 is ∆E1 = ( V12 + g z2 ) ρ Q ∆t. 2 Pressure P1 acting on the moving boundary a′ d′ produces work P1 A1 V1 ∆t = P1 Q ∆t, which should be added to the incoming energy E1 , and similarly ∆E2 = ( 1 2 b P∆ t a a’11111111 00000000 11111111 00000000 00000000 11111111 11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 ∆ 1 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 E P1 z1 d’ V1 V2 1111111111111111111111111111111 0000000000000000000000000000000 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 A 1111111111111111111111111111111 0000000000000000000000000000000 2 1111111111111111111111111111111 0000000000000000000000000000000 b’1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 2 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 P2 0000000000000000000000000000000 1111111111111111111111111111111 ∆ E2 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 c 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 c’ 1 V1 ∆ t A1 V2 ∆ t z2 Pf ∆ t d Fig. 1: the pressure work P2 Q ∆t should be added to the outgoing energy E2 . The energy of the fluid in the control volume will also be increased by the work of external forces P ∆t and dissipated due to the work of friction Pf ∆t. For a stationary flow the energy of the fluid inside the control volume does not change, and the total gain of energy is equal to the total energy loss: ρ V12 ρ V22 ( +ρ g z1 ) Q ∆t+P1 Q ∆t+P∆t = ( +ρ g z2 ) Q ∆t+P2 Q ∆t+Pf ∆t. 2 2 This gives the steady flow energy equation in the following form: P1 + P ρ V22 Pf ρ V12 + ρ g z1 + = P2 + + ρ g z2 + , 2 Q 2 Q (1) which represents conservation of energy per unit of fluid volume. It should be noted that the sign of the external power P is positive if the work done on the fluid (pumps, compressors) and negative when work done by the fluid (turbines). In engineering it is conventional to use the energy equation per unit of fluid weight, which can be obtained dividing (1) by ρg. The energy of fluid per unit of weight has dimensions of meters (Joules/Newtons) and is called head. The value P V2 H= + +z ρg 2g is called total head and represents the total energy of a unit weight of flowing fluid. It consists of pressure head, velocity head and potential head. The steady flow energy equation can now be formulated in terms of change of total head: P H2 − H1 = − hf , (2) ρgQ 3 z2 z1 r R U P1 d V(r) P2 x τ L Fig. 2: where hf = Pf /(ρ g Q) is head loss. Therefore, the total head of a steadily flowing fluid is increased by external work transferred to the unit of fluid weight, and decreased by the head loss due to viscous dissipation. Application to flow in a straight horizontal pipe The steady flow energy equations is widely applied in engineering for specifying flows of viscous fluid through pipes and pipe systems. In this section we consider a steady flow through a straight circular pipe of the internal radius R, diameter d (figure ). The flow in the pipe is parallel (this means that pressure is constant across the pipe), and the velocity profile does not change along the pipe. Such flow is called fully developed, and occurs far enough from the pipe inlet. The pipe wall is the boundary of a stream tube, and using one-dimensional approximation we take the mean velocity of the fluid through the pipe as the flow velocity. From the last term you should remember that the mean velocity U is a uniform velocity, which would provide the same flow rate Q through the pipe as the actual velocity profile V (r): Q = π R2 U = 2 π ZR 0 r V (r) dr. 4 Taking an average velocity as the flow velocity we make a mistake in determining the exact value of the kinetic energy of the flow because the mean of the square of velocity in the kinetic head does not equal to the square of the mean velocity. The factor by which the term U 2 /2g must be multiplied to get the exact value of the kinetic head is known as kinetic energy correction factor. Fortunately, for many practical flows this factor is close to 1. See the recommended literature for more details. For a pipe with constant cross-section area U is constant along the pipe. Then, for a horizontal pipe the steady flow energy equation (2) takes the form: P1 − P2 = ρ g hf . (3) For viscous fluid hf > 0, which means that steady flow of such fluid in a pipe is possible only if the pressure gradient is applied along the pipe. The head loss hf along the pipe can be conveniently measured by tube manometers. Referring to figure we can see that Pa = P1 − ρ g z1 = P2 − ρ g z2 , and hf = z1 − z2 . The head loss along the length L of the pipe is due to the friction on the pipe walls. We chose a cylindrical fluid particle as shown on figure . The particle is moving with a constant velocity, that is the total force acting on it is zero. This means that the pressure force on the vertical surfaces of the particle is balanced by the friction on the pipe wall. We can write: π d2 (P1 − P2 ) = π d L τw , (4) 4 where τ is the frictional shear stress on the pipe wall, which can be expressed in terms of the non-dimensional friction coefficient or friction factor of the pipe: τw f= . ρ U 2 /2 Substituting back into (3) and (4) we obtain the Darcy equation for head loss in a pipe: L U2 hf = 4 f . (5) d 2g Classical investigations of flows in pipes was performed by Osborn Reynolds who published his classical experimental results in 18831. In one of his experiments Reynolds studied the dependence of pressure gradient along a pipe 1 O.Reynolds (1883) An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and the law of resistance in parallel channels, Phil. Trans. Roy. Soc. 174, 935–82. Available online via JSTOR: http://www.jstor.org/view/03701662/ap000029/00a00170/0 5 Fig. 3: The experimental apparatus and results of experiments on flow in pipes from the original Reynolds paper. 6 from the pipe flow rate. The sketch of experimental apparatus and an example of the obtained results are illustrated on figure 3. Reynolds found the linear dependence for low flow rates, when the flow in the pipe is laminar. In logarithmic coordinates used on the figure this dependence is represented by the straight line with the slope 1. After the flow rate reaches some critical value rapid changes of flow characteristics occurs in the narrow region of flow rates (critical region), and the line inclination decreases. This new behaviour roughly corresponds to a power low with an exponent n < 1. For these larger values of flow rates flow in the pipe becomes turbulent leading to significant changes of flow characteristics. Reynolds found that the divergence from the laminar behaviour for a pipe flow always starts when the non-dimensional parameter known now as Reynolds number Ud , (6) ν reaches a certain critical value Rec . This value was found to be about 2300 and does not depends on pipe diameter and could be slightly lower for pipes with a rough surface. Extensive experiments for determination of friction factor f for circular pipes of different diameters and wall roughness carrying flows of various mean velocities had been carried by L.F.Moody in 1944. The results of these experiments are represented in the form of Moody diagram (figure 4), which is widely used for calculating flows trough pipes. It turns out that friction factor depends on the Reynolds number and on the relative roughness of the pipe wall k/d. Regions with different behaviour of f can be observed on the Moody diagram. For small values of the Reynolds number (laminar flows) the friction factor does not depends on the wall roughness, and is specified by a simple formula 16 f= . (7) Re In the narrow critical zone flow becomes turbulent and for larger Reynolds numbers f depends on both Re and k/d (transition zone). For very large Reynolds numbers (complete turbulence) f depends on k/d only. Moody diagram can be used to find the flow rate through a given pipe, when a given pressure difference is applied to pipe ends. For a laminar flow, using (3), (5), (6) and (7) we obtain the Poiseuille equation: Re = Q= π R4 ∆P , 8µ L where µ = ρ ν is dynamic viscosity. In the case of complete turbulence, keeping in mind that f does not depend on Re and therefore is constant for 7 Fig. 4: Moody diagram 8 a given pipe, we get: Q=π s R5 ∆P . fρ L Note different dependence of flow rate on pressure gradient in the pipe for laminar flow and complete turbulent flow. In the former case, the flow rate is proportional to pressure gradient, and in the latter case the flow rate proportional to the square root of pressure gradient. Head losses in pipe systems The steady flow energy equation and the theory of pipe flows with frictional losses considered on the previous lecture provide the basis for calculation of flows through systems of pipes and for designing of pipelines. The Darcy equation (5) can be used to specify the value of the head loss in a pipe which should be compensated by a certain amount of energy transferred to the fluid (pumping, height or pressure difference between the pipe ends) to provide the required mean velocity U and therefore the required flow rate through that pipe. In the most of practical cases the flow in pipelines is turbulent, when the friction coefficient f depends only on the relative wall roughness as can be seen from the Moody diagram (figure 4) Then the equation (5) for each pipe in a pipeline can be rewritten as: hf = Kf U2 2g (8) where the loss coefficient Kf depends only on the properties of a particular pipe in the pipeline and does not depend on the flow rate. Pipes are not the only elements of pipelines where head loss is possible. The head loss can also occur in pipe fittings, bends, contractions, etc. Flow rate through a pipeline can be regulated by changing a head loss in valves. Equation (8) provides the general form of equations used to calculate head loss in various elements of a pipeline, with a specific empirical (that is found from an experiment) loss coefficient Kf for each such an element. Examples of pipe line elements with loss coefficients can be found in the literature. Pipes in series If pipes or other elements are connected in series, that is from end to end, the total head loss is the sum of losses in all individual elements. It is convenient to express equations (5) and (8) for each element by using the flow rate Q = A U, which is the same for all elements connected in series. Then the 9 1 hf . 2 z2 z1 . Fig. 5: head loss for an individual element i can be written as hi = κi Q2 , with a suitable coefficient κi for each element. Then the head loss of the entire system is X hf = Q2 κi . i Example: Water flows between tanks with water levels z1 and z2 trough two identical pipes of length l and diameter d (figure 5). The pipes are connected by an elbow with the loss coefficient K1 = 0.1. Sharp edged inlet and outlet have the loss coefficients K2 = 0.5 and K3 = 1 respectively. The friction coefficient f of the pipes is constant for given flow conditions. Find the flow rate of fluid in the pipes. For a steady flow the head loss along the path 1–2 should be balanced by the difference of the total head at points 1 and 2. Pressure at points 1 and 2 is the same, and velocities there are negligibly small. Therefore, only the potential head contributes to the total head difference. That is hf = z1 − z2 . The head loss along the pipes is the sum of individual losses 2 l l U hf = 4 f + 4 f + K1 + K2 + K3 , d d 2g and the corresponding flow rate is s 2 πd 2 g ( z1 − z2 ) Q= . 4 4 f (l/d) + K1 + K2 + K3 10 B C 1 2 A Fig. 6: Parallel pipes The flow divides between two or more pipes and then comes together again. For such pipe systems the sum of flow rates through individual components is the entire flow rate through the system X Q= Qi . i For each element of the parallel pipe system the difference of the total head between its ends is the same, which means that all elements have the same head loss κi Q2i = hh . For an N-element system (i = 1, 2, 3, . . . N) we usually have an unknown head loss hf and N unknown flow rates Qi . To find these N + 1 values we can use N + 1 equations above. Example: Two identical pipes A and B have length L and cross section area A. The pipes are connected in parallel to a pipeline with a constant flow rate Q (figure 6). A valve C is used to regulate flows through the pipes. When the valve is fully opened its loss coefficient is K = 0.3. Friction coefficient f of the pipes is constant under the given flow conditions, and the losses in fittings are negligible. Find the minimal and the maximal flow rate trough pipe A. Head loss in each pipe between 1 and 2 is equal to the difference in the total heads between these points: H1 − H2 = hA = hB . 11 Head losses in A and B are: hA,B = 4 f L Q2A L Q2B = (4 f + K) d 2 A2 g d 2 A2 g and the sum of flow rates in A and B is the total flow rate of the system: QA + QB = Q. For the flow rate QA we obtain the following quadratic equation Q2A − 2 Q(1 + 4f L 4f L ) QA + Q2 (1 + )=0 dK dK with solutions QA = Q 4f L ± 1+ dK r 4f L 4f L (1 + ) dK dK ! . (9) The flow rate QA should be less then Q, therefore we should take the solution with the minus sign. Maximal QA corresponds to the closed valve (K = ∞), when all fluid flows through the pipe A: QA = Q. The minimal QA corresponds to the fully opened valve, and its value can be found by substituting the value of K into the formula (9). Note, that if K = 0 (no extra valve resistance) both parallel branches are identical, and have the same flow rate QA = QB = Q/2. Equation (9) gives value QA = Q/2 in the limit K → 0. Pipe branches A classical problem with a branching pipeline is the three reservoir problem illustrated on figure 7. Three reservoirs with different water levels are connected by three pipes with a junction point 0 and unknown flow rates. One of the difficulties of the problem is that we usually do not know the flow direction in one of the branches (branch 3) before solving the problem. General principles applied for solving the three reservoir problem and other problems with branching pipes are: 1. The uniqueness of the total head. This means that at each point of a pipeline the total head have only one value. The important consequence of this property is that the value of the total head at a junction point is the same for all pipes. 2. The continuity principle is applied to a junction point. That is the flow into the junction is the same as the flow from the junction. 12 1 . 3 . Q1 Q3 z1 0 z3 Q2 2 z2 z0 . Fig. 7: 3. Darcy’s equation is applied to each pipe with additional losses in fittings. Note, that for long pipelines the frictional losses in pipes provide the major contribution to the total head loss and minor losses in fitting can often be neglected. As before, we represent the head loss in each branch in the form hi = κi Q2i . Assuming originally the direction of the flow in the pipe 0–3 as shown on the figure 7 and using the principles stated above we can write: z1 − H0 = κ1 Q21 H0 − z2 = κ1 Q22 H0 − z3 = κ1 Q23 Q1 = Q2 + Q3 . Thus, we have 4 equations to specify three unknown flow rates Q1 , Q2 , Q3 , and the unknown total head H0 at the junction. The algebraic solution of these equations is tedious and not possible for more than 3 pipes. However, we can use the trial and error method, taking some value of H0 as an initial guess, calculating the flow rates and then checking the continuity condition. 13 . H0 Q ∆H Ps . Fig. 8: I is useful to use trial points to plot the graph of Q1 − Q2 − Q3 as function of H0 . The required solution will be an intersection of this graph with the horizontal axis. If the original choice of the flow direction in the pipe 0–3 is wrong, the solution will not be possible. For an opposite flow in 0–3 the equations become: z1 − H0 = κ1 Q21 H0 − z2 = κ1 Q22 z3 − H0 = κ1 Q23 Q1 + Q3 = Q2 . The two systems of equations become identical if H0 = z3 in which case Q3 = 0. It is convenient to chose H0 = z3 as an initial guess to specify the flow direction in 0–3. If Q1 > Q2 , then the flow is from 0 to 3, and if Q2 > Q1 , the flow is from 3 to 0. Pumping A pump can can deliver water to a higher level by transmitting energy to the flow. Pumps are characterised by the total head ∆H applied to the fluid, discharge Q and the shaft horsepower Ps . A particular pump can provide a head increase ∆H for a discharge Q and requires power Ps on the shaft. The curve expressing the relationship between pump discharge and head is called the characteristic curve or head curve, and the curve specifying the required power is the power curve. A typical example of pump characteristics is given κ1 κ2 H0 14 c BJM Pumps http://www.bjmpumps.com Fig. 9: Typical pump characteristics. 15 on figure 9, where B.H.P. stands for brake horsepower, that is the power of an engine driving the pump shaft. The power transmitted by a pump to water flow or water horsepower is specified as Pw = ρ g Q ∆H, and the pump efficiency is η = Pw /Ps , that is the ratio of utilised power to the total power required by a pump, and the value of η is always less then 1. To specify the working regime of a pump in a pipeline we have to equate the head provided by the pump at a specific flow rate with the pumping head H0 (figure 8) and head losses in the pipeline for that flow rate: ∆H = H0 + κ Q2 , where the value of κ can be regulated by the valve, which will change the flow in the pipeline. The problem can be solved graphically, by plotting the parabola of the required head H0 + κ Q2 on the pump diagram and finding its intersection with the characteristic curve of the pump. Quasi-steady flows The energy equation have been derived for the case of a steady flow. For a flow in a pipeline this means that flow rates and heads at different points of the pipeline do not depend on time. However, if flow changes very slowly and unsteady effects can be neglected, we can apply the steady flow energy equation with sufficient accuracy at any instant during the process. Unsteady flows with slowly changing parameters which can be assumed steady at each time moment are called quasi-steady flows. For example, if a tank with a large area of water surface A (figure 10) is drained through a pipe of a much smaller cross section a the water level in the tank will change very slowly and we can calculate the flow rate Q through the pipe at each time instant t by taking the current value of the water level Z(t) and applying the steady flow energy equation as if Z was constant: H1 − H3 = K U2 , 2g where the total heads of the water surface in the tank of the jet at the pipe outlet are Pa U2 Pa H1 = Z + and H3 = + ρg 2g ρg 16 A 1 . Z(t) Z1 2 Z2 Q ( t ) = − A dZ / dt a 3 Fig. 10: respectively, and K is the total loss coefficient of the pipeline including friction losses in the pipe, fittings losses, entry losses, etc. This gives Z(t) = (1 + K) Q2 , 2 a2 g and using the relation between the flow rate and the water level Q = −A dZ/dt we obtain the following differential equation describing the evolution of the water level in the tank: r dZ a 2g p =− Z(t) . dt A 1+K We can see that when the value of the area ratio a/A is small and the water level Z(t) is not too big, then the non-stationary term dZ/dt is small and the application of the steady flow energy equation is justified. This equation can be used to calculate the time required to change the level of water in the tank from the initial level Z(0) = z1 to any given level Z(t) = z: s s Zz √ A 1+K dZ A 1+K √ √ = t=− ( z1 − z ) . a 2g a 2g Z z1 This equation can also be used to determine the level z left after a given time t has elapsed.