Fluid Mechanics & Hydraulics Review Notes PDF

## Fluid Mechanics & Hydraulics Review Notes ### Chapter 1: Properties of Fluids | Concept | Formula | Notes | | --------------- | ------------------ | --------------------------------------------------------------------------------- | | Unit Weight | Y = W/V = pg | | | Mass Density | p = M/V | Rair = 287 J/kg-°K | | Specific Volume | Vs = 1/p | | | Specific Gravity | Sliquid = Yliquid/Ywater, Sgas = Ygas/Yair | Pwater = 1000 kg/m³ Ywater = 9810 N/m³ Pair = 1.225 kg/m³ Yair = 12 N/m³ | | Dynamic Viscosity | τ = F/A, μ = dV/dy = U/y | 0.1 Pas = 1 poise | | Kinematic Viscosity | v = μ/p | 0.0001 m²/s = 1 stoke | | Droplet Pressure | p = 4σ/d | | | Capillarity | h = 4σcos(θ)/dy | | | Compressibility | β = AV/VΔp | | | Bulk Modulus of Elasticity | 1/β = Eв = 1/β | | | Celerity | c = √Eв/p | | | Gas Law | P₁V₁ = P₂V₂ | T°K = °C + 273 | | Gas Law (Adiabatic, Isentropic) | P₁V₁ᵏ = P₂V₂ᵏ, T₂/T₁ = (P₂/P₁)ᵏ⁻¹ | k = adiabatic exponent | ### Chapter 2: Principles of Hydrostatics | Concept | Formula | Notes | | --------------- | --------------------- | ---------------------------------------------------------------------------------------- | | Pressure | p = F/A | Acts normal to the area | | Pascal's Law | | Pressure on a fluid is equal in all directions and in all parts of the container | | Gage Pressure | | pressures above or below the atmosphere and can be measured by pressure gauges or manometer | | Atmospheric Pressure | Patm = 101.325 kPa = 14.7 psi | | | Absolute Pressure | Pabs = Patm + Pgage | | | Fluid Pressure | p = yh | Any change in pressure at point A would cause an equal change at another point B If a point lies on the free liquid surface, then the gage pressure at that point is zero | | Pressure Below Layers of Fluids | p = ∑ynhn | If two points lie on the same elevation, then their pressure is also the same | ### Chapter 3: Total Hydrostatic Force on Surfaces | Concept | Formula | Notes | | --------------- | --------------------------- | ----------------------------------------------------------------------------------------------- | | Force due to Pressure | F = pA | Acts normal to the area | | Hydrostatic Force on Inclined Surfaces | F = yhA or F = PcgA, e = Icg/Ay | Fh = уҺА ог FH = PcgA Fy = yV "Any body immersed in a fluid is acted upon by an upward force (buoyant force) equal to the weight of the displaced fluid" | | Hydrostatic Force on Curved Surfaces | | | | Archimedes’ Principle | | BF = YVD | | Buoyant Force | | Buoyant Force only happens when liquid is also present under a body | | Stability of Floating Bodies | MG MB GB, MB'= 1+1/2D/I * tan² (θ) RM or OM = Wx = W(MGsine) | +MG = stable, MG = unstable | ### Chapter 4: Relative Equilibrium of Liquids | Concept | Formula | Notes | | --------------- | ------------------------------------------- | ------------------------------------------------------------------------------------------------------------ | | Horizontal Rectilinear Translation | REF = ma, N = W = mg, tan θ = a/g | Fluid mass in horizontal accelerated motion | | Inclined Rectilinear Motion | REF = ma, W = mg, tan θ = ax/(g±ay) | | | Vertical Rectilinear Motion | REF = ma, W = V, Area = A, F = PA, p = yh(1 ± rω²/g) | | | Rotating Motion | tan θ = ω²x/g, y = ω²x²/2g | For Closed Cylindrical Vessel: When y > H²/2D, Depth of Water in the Tank when Rotation Stop: H²/(2y) | ### Chapter 5: Fundamentals of Fluid Flow | Concept | Formula | Notes | | --------------- | --------------------------------------------------- | -------------------------------------------------------------------------------------------- | | Discharge or Flow Rate | Q = Av, M = pQ, W = YQ | Q = volume flow rate M = mass flow rate W = weight flow rate | | Continuity Equation | Q = A₁V₁ = A₂V₂ = constant, Q = P₁A₁V₁ = P₂A₂V₂ = constant, Q = Y₁A₁V₁ = Y₂A₂V₂ = constant | incompressible Fluids: Compressible Fluids: | | Bernoulli’s Energy Theorem | E₁ = E₂, Vp₁/2g + P₁/Y + Z₁ = Vp₂/2g + P₂/Y + Z₂ | v²/2g = kinetic head P/Y = pressure head z = elevation head | | Power and Efficiency | P = YQE, Eff = Output/Input | | | Energy Head Gains and Losses | E₁ + Hp - HF - H₁ = E₂ | Hp = pump head gain HF = friction head loss H₁ = turbine head loss | ### Chapter 6: Fluid Flow Measurement | Concept | Formula | Notes | | ------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Device Coefficients | C = Q/QT, Cc = A/AT, Cv = V/VT, HL = 2*(1 - Cc)*(1 - Cc) * v²/2g | Coefficient = Actual/Theoretical C = CcCv When A₁ is significantly greater than A₂ make (1/A₂) = 1 H = total head producing flow PA - PB/Y + v²/2g Section at which the contraction of the je ceases | | Orifice Velocity | v = √2gH | | | Vena Contrata | Occurs at from the upstream face | | | Orifices under Low Heads | Q = C√2gL[H₁³/² - H₂³/²] | From: Q = CAv dQ = CdAv → dQ = CdA√2gH | | Unsteady Flow Discharge Time | t = 2As/(CA√2g) | From: V = Qdt dV = dQdt → dt = dV/dQ | | Unsteady Flow Two Connected Tanks Discharge Time | t = 2AHS(H₁ - H₂)/(CA√2g) | AsdH/dt = CA√2gH → t = H₁/CA√2g H2/ CA√2g, VH = As1As2/(As1 + As2) | | Weirs | Discharge General Formula: dQ = CdA√2gH → Q = H₁/H2 * CA(H)/2gHdH Q = C√2g•LH³/² = CWLH³/² Q = 1.84LH³/² C√2g LH³/² = Cw tan(θ/2)H⁵/² Q = 4/15 * C√2g LH³/² = Cw tan(θ/2) H⁵/² | From: Q = CAv → Q = CA√2gH Express area A as a function of head H Cw = C√2g For Contracted Weirs: L=L-0.1H → Singly Contracted, Le L-0.2H → Doubly Contracted Considering Velocity of Approach: use H+v²/2g as H³/² L = 2 Htan(θ/2) Cw = C√2g Cipolletti Weirs have 1H:4V slopes From: V = Q/t →t = H₁/H2 * A√dH/Q Express V and Q as a function of head H | | Unsteady Flow in Weirs | t = H₁/H2 * AdH/Q | | ### Chapter 7: Fluid Flow in Pipes | Concept | Formula | Notes | | --------------- | -------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | Reynold's Number | Re = vD/μ = vDρ/μ | v = mean velocity v = kinematic viscosity ρ = dynamic viscosity D = pipe diameter For non-circular pipes: D = 4R From other sources: Transition to Turbulence is at Re = 2300 | | Flow Type | Re < 2000 → Laminar Re > 2000 → Turbulent Re = 2000 → Critical Velocity, For Laminar Flow: f = 64/Re | | | Friction Factor Values | For Turbulent Flow: f = 0.25log(ɛ/D/3.7 + 5.74/Re⁰.⁹)² For Laminar Flow: f = 64/Re For Turbulent Flow: Le = 4.4DR¹/⁶ For Laminar Flow: Le = 0.06DRe | ɛ = absolute roughness ɛ/D = relative roughness | | Entrance Length | | | | Velocity Distributions in Pipes | u = Vc (1 - (r/R)²) For Turbulent Flow: v = vc - 3.75 * (vc√f)log(R/(R-r)) vc = v(1 + 1.33√f) u = vc - 5.75 (yc/ρL) * log(R/(R-r)) | v = average velocity vc = max velocity u = velocity at distance r | | Shearing Stress in Pipes | τs = yh, τs = yh/2L | Max shearing stress is at pipe walls: r = R | | Major Head Loss: Darcy-Wiesbach | General: hf = fLv²/D * 2g, hf = 0.0826fLQ²/D⁵ | Derived from: V = -R²S² hf = L/A × S² | | Major Head Loss: Manning | General: hf = 6.35n²Lv²/D³, hf = 10.29n²LQ²/D³ | hf = L/A × S² | | Major Head Loss: Hazen Williams | General: hf = 1.354LQ¹.⁸⁵/C₁.⁸⁵R¹.₁⁶⁷A¹.⁸⁵, hf = 10.67LQ¹.⁸⁵/C₁.⁸⁵D⁴.⁸⁷ | Derived from: v = 0.849C₁R⁰.⁶³ S⁰.⁵⁴ | | Minor Head Losses | hf = kmv²/2g | km = minor hf coefficient | ### Chapter 8: Open Channel | Concept | Formula | Notes | | --------------- | --------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Specific Energy | H = v²/2g + d | Energy per unit weight relative to the bottom of the channel A = area R = hydraulic radius = A/Pw | | Chézy Formula | v = CR¹/²S¹/², Q = ACR¹/²S¹/² | C = Chézy coefficient S = slope = h/L | | Chézy Coefficients | Kutter and Ganguillet Formula: C = 1/n + 23 + 0.00155/S + (23 + 0.00155)/(1 + √S) Manning Formula: C = 1/n * R¹⁶ Bazin Formula: C = 87/(1 + m/√R) Powell Formula: C = -42 log(n/√R) | n = roughness coefficient m = Bazin coefficient R = hydraulic radius ɛ = roughness Re = Reynolds number S = slope of energy grade line | | Uniform Flow | S = So (slope of channel bed) | The velocity, depth of flow, and cross-sectional area of flow at any point of the stream is constant. | | Boundary Shear Stress | τo = yRS | | | Normal Depth | dn occurs when S = So | Note that A, n, and S are constant, therefore, to maximize Q, R must be maximized. R can be maximized when Pw is minimum for a given A. QxR → Q&A/Pw | | Most Efficient Cross Sections | Using Chézy-Manning Formula: v = 1/n * R²/³S¹/² Q = A * R²/³S¹/² Rectangular Section: T = 2d, R= d/2 Triangular Section: T = 2d, θ = 90°, R = d/2√2 Trapezoidal Section: T = 2s, R = d/2 Circular Section: Qmax = occurs when d = 0.938D, Vmax = occurs when d = 0.820D | T = top width d = depth s = sides | | Velocity Distribution in Open Channel | u= v + √gys(1 + 2.3log(y/h)) | v = depth of flow y = depth of water in channel u = velocity at distance y'from bed K = Kármán constant v = mean velocity of flow S = slope of EGL | ### Chapter 9: Hydrodynamics | Concept | Formula | Notes | | ------------------- | ------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------ | | Reaction Against Flat Plates | For Fixed Plates: R=pQv For Moving Plates: R=pQ'u | F=ma =(pv)(2-1)/t p = density of fluid Q = discharge v = velocity of liquid v' = velocity of plate u = relative velocity = v - v' Q' = relative discharge = Au | | Force Against Vanes | For Fixed Vanes: Fx = pQ(V₁x-V₂x), Fy=pQ(V₁y-V₂y), F=√Fx²+Fy², Ф = tan⁻¹(Fy/Fx) For Moving Vanes: Fx = pQ'(V₁x - V₂x), Fy = pQ'(V₁y - V₂y), F =√Fx²+Fy², Ф = tan⁻¹(Fy/Fx) | V₂x = v' + ucosθ p = density of fluid Q = discharge v = velocity of liquid v' = velocity of plate u = relative velocity = v - v' Q' = relative discharge = Au For Fixed Vanes: F = 2pAv² sin θ For Moving Vanes: Fy = pAu²(1 - cos θ), F₂y = -pAu² sin θ | | Forces Developed in Pipes | Fx = ΣpQ(V₂x - V₁x), Fy = ΣpQ(V₂y - V₁y), R=√Rx²+Ry², Ф = tan⁻¹(Ry/Rx) | F = pA | ### Chapter 10: Deep Foundations (Cohesive Soil) | Concept | Formula | Notes | | ------------------- | ----------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Ultimate Capacity of a Single Pile | Ultimate Bearing Capacity of Pile | Qb = CNcAtip = 9cAtip | N = 9 c = cohesion of soil Atip = cross-sectional area of pile | | Ultimate Frictional Capacity of Pile (a method) | Qf = cLaP | c = cohesion of soil L = length of pile a = adhesion factor (1 if not given) P = perimeter of pile | | Ultimate Frictional Capacity of Pile (β method) | Qf = PLẞσε | P = perimeter of pile L = length of pile β = frictional coefficient For normally consolidated clay: β = (1 - sin φ)tan φ For over consolidated clay: β = (1 - sin φ)tan φVOCR | | Ultimate Frictional Capacity of Pile (λ method) | Qf = PLA(σε + qu) | P = perimeter of pile L = length of pile λ = frictional coefficient qu = unconfined compression strength qu= Cu/2 Cu = unconfined shear strength/cohesion σε = effective/average stress at mid - height of the clay | | Ultimate Capacity of Pile | Qult = Qb + Qf | | | Allowable Capacity of Pile | Qall = Qult / FS | | | Individual Action | Qult = (Qb + QF) × n | | | Block Action | Qult = (Qb(block) + Qf(block)) × n | | | Group Action | Qult = (Qb + QF) × n | | | Ultimate Capacity of Group Piles | Qall = Qall(single) × Eff, Eff = 2(m + n²/2)S + 4D, Eeff = πDmn | m = number of pile columns n = number of pile rows S = spacing of pile D = diameter of pile | ### Chapter 11: Slope Stability | Concept | Formula | Notes | | ------------------- | ----------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------ | | Infinite Slope (Clay Soil) | No Water Pressure: FS = C / tan φ + γH cos² βtan β / tan φ With Water Pressure: FS = C / Yefftan φ + YsatH cos² βtan β / Ysat tan φ | FS = factor of safety c = cohesion of soil H = height of soil β = angle that the slope makes with the horizontal φ = angle of friction of soil y = unit weight of soil Ysat = saturated unit weight of soil Yeff = ysat-yw | | Infinite Slope (Sandy Soil) | Without Seepage: FS = tan φ / tan β With Partial Seepage: FS = (1 - Ywh / YsatH) tan φ / tan β With Full Seepage: FS = Yeff tan φ / Ysattan β | h = height of water seepage from bottom sand stratum |

Fluid Mechanics & Hydraulics Review Notes PDF

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