The standard deviation σ is used for the index of fluctuation.

Transcription

1 Chapter 2: Unit and Measurement 2.1 Unit and Significant Figures Unit The relationship between the Celsius (C) and Fahrenheit (F) of temperature is represented by the equation 2.1. F=(9/5)C+32 (2.1) 2.2 Measurement Non-linear and equilibrium Wheatstone bridge (Fig. 2.11), R2/R1=R3/Rx (2.2) When the three resistances R1, R2, R3 are known, the unknown resistance Rx is calculated Noise and Statistics In order to correct the errors due to fluctuation, μ is used as the mean value of the values xk measured N times (from k=1 to k=n). N μ = 1 (X N k=1 k) (2.3) The standard deviation σ is used for the index of fluctuation. σ = N k=1 (X k μ) 2 N (2.4) Chapter 3: Materials 3.1 Deformation Stress-strain diagram The ratio of the deformed length Δx against the original length x is called strain ε. ε=δx/x (3.1) When the material is pulled in one direction, the ratio of elongation at the parallel direction of the force application (strain Δx) to contraction at the vertical direction (strain Δy) is Poisson's ratio κ.

2 κ=δy/δx (3.2) Consider about the following deformation: the quadrangular prism with the height x at the side of the bottom square y, while maintaining its volume, deforms to the quadrangular prism with the height x + Δx at the side of the bottom square y-δy (Fig. 3.9). (x+δx)(y-δy) 2 =x y 2 (3.3) (x+δx)(y 2-2 y Δy+(Δy) 2 ) =x y 2 +y 2 Δx-2 x y Δy-2 y Δx Δy+x (Δy) 2 +Δx (Δy) 2 =x y 2 (3.4) y 2 Δx-2 x y Δy=0 (3.5) x 2 Δx-2 x 2 Δy=0 (3.6) Δy/Δx=1/2 (3.7) The cross-sectional area of the vertical direction decreases from A0 to A1. True stress: τ1 Nominal stress: τ0 τ1=f1/a1 τ0=f1/a0 (3.8) (3.9) Sphere Radius: r The inner and outer pressure difference: ΔP Surface tension: γ (Fig. 3.14) ΔP π r 2 =2π r γ (3.10) ΔP=2γ/r (3.11) Bending The x-axis is defined along the bar (the length of a) from the left end to the right in Fig The upward force is defined as positive. Eq describes the balance of the forces F.

3 F/2+F/2-F =0 (3.12) -(a/2)f+a(f/2)=0 (3.13) The counter-clockwise moment is defined as positive in Fig Eq describes the balance of moment M around the left edge of the bar. M-x (F/2)=0 (3.14) The moment of inertia of area A is defined as Iz in Eq The very small area: da The distance y from the z-axis Iz= y 2 da (3.15) Chapter 4: Flow 4.1 Fluid and solid Fluid and pressure A: the flow passage cross-sectional area [m 2 ], v : the flow velocity [m s -1 ]. A1 v1=a2 v2 (4.1) In equation (4.2), ρ is the density [kg m -3 ], v is the flow rate [m s -1 ], and p is the pressure [Pa]. (1/2) ρ v 2 +p =constant (4.2) In the gravitational field, fluid pressure p [Pa] is generated as the product of the density ρ [kg m -3 ], the gravitational acceleration g [m s -2 ] and the height h [m] (Fig. 4.5). p=ρ g h (4.3) In the gravitational field, the water of 10 m height generates the following pressure: kg m m s m=98 kpa (4.4) The water pressure at the point elevated by 10 m under 1 atm ( kpa) is kpa-98 kpa=3.325 kpa (4.5) The vapor pressure of water is kpa at 298 K Elasticity and viscosity The shear rate (γ) is the quotient of the speed difference between the surfaces (Δv)

4 divided by the distance (y) between the surfaces (Fig. 4.10). γ=δv/y (4.6) Coefficient of viscosity: η Shear stress: τ η=τ/γ (4.7) 1 P=1 dyn cm -2 s=0.1 N m -2 s=0.1 Pa s (4.8) 4.2 Resistance of flow and distribution of velocity Resistance of flow The flow rate Q [m 3 s -1 ] The difference of pressure between upstream and downstream: ΔP [Pa]. The flow resistance: Rf [Pa m -3 s]. Rf=ΔP/Q (4.9) The electric resistance: R The difference of electric potential: ΔE The electric current: I R=ΔE/I (4.10) Systemic circulation resistance (total peripheral resistance) Rs is a quotient of the difference between aortic pressure Pa and right atrial pressure Pr divided by cardiac output Qc (Fig. 4.19). Rs=(Pa-Pr)/Qc (4.11) Pulmonary circulation resistance (Rp) is a quotient of the difference between pulmonary artery pressure (Pp) and left atrial pressure (Pl) divided by cardiac output (Qc). Rp=(Pp-Pl)/Qc (4.12) Resistivity: ρ [Ω m] Electrical resistance R [Ω] of the metal wire is proportional to the length l [m], and is inversely proportional to the cross-sectional area A [m 2 ]. R=ρl/A (4.13) Hagen-Poiseuille flow

5 Fig shows concentric thin cylinders (length l, velocity v, radius r, and thickness dr) in the circular tube (radius a). F is the product of the lateral area of the cylinder (2 π r l) and the shear stress τ. F=2 π r l τ (4.14) τ=-η dv/dr (4.15) where η is coefficient of viscosity, dv / dr is the shear rate. The origin of r is the center of the circular tube. The velocity v has the maximum value at the center (r = 0). F=-2 π l η r dv/dr (4.16) df=-2 π l η(1/dr)(r dv/dr) dr (4.17) Pressure difference: ΔP df=2 π r dr ΔP (4.18) -2 π l η(1/dr)(r dv/dr) dr=2 π r dr ΔP (4.19) -(1/dr)(r dv/dr) dr=(δp/(l η))r dr (4.20) - (1/dr)(r dv/dr) dr=(δp/(l η)) r dr (4.21) Integration constant: C -r dv/dr=(δp/(l η))(r 2 /2)+C (4.22) r dv/dr=-(δp/(l η))(r 2 /2) (4.23) dv/dr=-(δp/(l η))(r/2) (4.24) The shear rate at the wall: γw γw=-(δp/(l η))(a/2) (4.25) dv=-(δp/(l η)) (r/2) dr (4.26) Integration constant: D v=-(δp/(4 l η))r 2 +D (4.27) D=(ΔP/(4 l η))a 2 (4.28) v=(δp/(4 l η))(a 2 -r 2 ) (4.29) Flow rate: Q a Q = 2πr dr ΔP 4l η (a2 r 2 ) 0 π ΔP Q = 2l η (a2 r r 3 )dr Q = 0 a a π ΔP 2l η [a2 r 2 2 r4 4 ] 0 (4.30) (4.31) (4.32)

6 Q=πa 4 ΔP/(8 l η) (4.33) ΔP=8 l η Q/(πa 4 ) (4.34) γw=-4 Q/(πa 3 ) (4.35) Flow resistance: Rf Rf=(8 l η)/(πa 4 ) (4.36) Couette Flow: In the Couette flow, the shear rate γ is constant regardless of the distance from the wall. Velocity: v Distance between walls: d γ=v/d (4.37) θ is the clearance angle [rad]. The speed v [m s -1 ] of the conical surface is proportional to the distance r [m] from the axis of rotation. d=r tan θ=r θ (4.38) ω is the angular velocity [rad s -1 ]. The shear rate γ is constant regardless of the distance r from the axis of rotation. v=r ω (4.39) γ=v/d=r ω/(r θ)=ω/θ (4.40) When the torque increases from T0 to T1, the clotting ratio Rc is calculated by using equation (4.41) (Fig (b)) Rc=(T1-T0)/T1 (4.41) Flow between parallel walls The origin is defined at the center between the parallel walls. The y-axis is defined vertically towards the wall. A virtual thin flat plate sandwiched between y and y + dy has the velocity of v, the width of b, and the length of l (Fig (b)). When the plate has uniform linear motion, the friction force ΔF between the plates balances with the force by the pressure difference ΔP between the upstream and the downstream. The friction force F is the product of the friction area of the plate (b l) and the shear stress τ. F=b l τ (4.42)

7 τ=-η dv/dy (4.43) In formula (4. 43), η is the viscosity coefficient of the fluid. At the center (y = 0), v is maximum. The right-hand side of the equation (4.43) has the negative sign, because the velocity v decreases with y. F=-b l η dv/dy (4.44) df=-b l η(1/dy)(dv/dy)dy (4.45) df balances with the force at the end face of the thin flat plate (area of b dy) by the pressure difference ΔP between the upstream and the downstream. df=b dy ΔP (4.46) -b l η(1/dy)(dv/dy)dy=b dy ΔP (4.47) -(1/dy)(dv/dy)dy=(ΔP/(l η))dy (4.48) Both sides are integrated by y, - (1/dy)(dv/dy)dy=(ΔP/(l η)) dy (4.49) -dv/dy=(δp/(l η))y+c (4.50) C is an integration constant. dv/dy=-(δp/(l η))y (4.51) Shear rate on the wall (γw) at y= d / 2 is, γw=-(δp/(l η))(d/2) (4.52) dv=-(δp/(l η))y dy (4.53) dv=-(δp/(l η)) y dy (4.54) v=-(δp/(2 l η))y 2 +D (4.55) D is a constant of integration. The Flow velocity is zero on the wall (v=0 at y=d/ 2). D=(ΔP/(8 l η))d 2 (4.56) v=(δp/(8 l η))(d 2-4y 2 ) (4.57) d 2 Q = 2b ΔP 8l η (d2 4y 2 )dy 0 (4.58) Flow rate: Q Q = b ΔP 4l η [d2 y 4 y3 d 2 3 ] 0 (4.59)

8 Q=b d 3 ΔP/(12 l η) (4.60) ΔP=12 l η Q/(b d 3 ) (4.61) γw=6 Q /(b d 2 ) (4.62) Secondary flow Method of measuring the viscosity (η) of the surrounding fluid by using the falling ball: Velocity of the ball: v Diameter: d The difference between the density of the ball (sphere) and the density of the surrounding fluid: (ρ1-ρ2 ) η d 2 (ρ1-ρ2 )/v (4.63) Laminar flow and turbulent flow Reynolds number (Re) is calculated by the equation (4.64), where ρ is the density, v is the representative velocity, x is the representative length, and η is the viscosity coefficient. Re=ρ v x/η (4.64) Chapter 5: Energy 5.1 State of substance Temperature Temperature T Volume V Pressure P n moles of ideal gas R is the gas constant of 8.3 J K -1 mol -1. P=(n/V) R T (5.1) Hydrogen ion concentration index Power of hydrogen: ph [H + ] is concentration of hydrogen ion.

9 ph=-log10[h + ] (5.2) The chemical equilibrium controls carbon dioxide and hydrogen ion through carbonate. H + +HCO - 3 H 2 CO 3 H 2 O+CO 2 (5.3) Heat Enthalpy: H H=U+PV (5.4) In equation (5.4), U is the internal energy, P is pressure, and V is the volume. In a system, entropy increases (ds) [J K -1 ], when the system has received amount of heat δq [J] from the heat source at temperature T [K]. ds =δq/t (5.5) Heat transfer coefficient: Hc Heat transfer Q [J] is proportional to the area A [m 2 ], time t [s], and the temperature difference ΔT [K]. Q=Hc A t ΔT (5.6) 5.3 Substance transportation Permeability through membrane The permeate flow rate of the gas (Q) through the membrane is proportional to the gas partial pressure difference ΔP and the membrane area S, and inversely proportional to the thickness of the membrane d (Fig. 5.9). Q S ΔP/d (5.7) Osmotic pressure The osmotic pressure: P P=(n/V) R T (5.8) In the equation (5.8), (see equation (5.1)) n / V is the molar concentration of the solution [mol m -3 ], R is the gas constant, and T is the temperature [K]. Chapter 6: Movement

10 6.1 Balance among forces and control of movement Balance among forces See Fig. 6.3(a). In equation (6.1), m is the mass [kg], v is the speed [m s -1 ], r is the radius [m], and ω is the angular velocity [rad s -1 ]. F=m v 2 /r =m r ω 2 (6.1) The deformation ratio y is calculated by the equation (6.2), where a is the length of the major axis of the ellipsoid, b is the length of the minor axis of the ellipsoid (Fig. 6.4 (a)). The value of y is zero at a sphere (a = b), and approaches to unity as the deformation of the ellipsoid advances (a >> b). y=(a-b)/(a+b) (6.2) The maximum deformation ratio: y 0 The characteristic stress: τ0 e 2.72 y = y 0 (1 e τ τ0 ) (6.3) In formula (6.3), τ is stress. τ0 is the stress, when y reaches to the 63% of y Lubrication and wear Machine elements and systems In Eq. 6.4, y0 is the saturation value of the displacement, t is the time, and e is the base of natural logarithms (e 2.72). The time constant t0 corresponds to the time, when y reaches to 63% of the saturation value (see Equation (6.3)). y = y 0 (1 e t t0 ) (6.4) Coefficient of friction The ratio of the friction force (Fs) and the normal force (Fn) is called the coefficient of friction (μ). μ=fs/fn (6.5) θ is called friction angle (Fig. 6.13). μ0 is called the maximum coefficient of static friction. (M g sin θ0)/( M g cos θ0)=tan θ0=μ0 (6.6)

11 In equation (6.6), M is the mass of the object, and g is the gravitational acceleration. Chapter 7: Designing and Machining Surface roughness The arithmetic average roughness: R a (Fig. 7.6) The extraction line segment: l The deviation (y(x)) from the average surface R a = 1 l l y(x) dx (7.1) 0 The ten-point average roughness Rz (Fig. 7.8) at the extracted section of line l is the average value of the difference from the summit to the bottom. The data (y) of the height of the summit and the depth of the valley from the mean plane are used. The sum of the five absolute values of the height (from the maximum to the fifth) and five absolute values of the depth (from the maximum to the fifth) is divided by five. R z = y 1 + y 2 + y 3 + y 4 + y 5 + y 6 + y 7 + y 8 + y 9 + y 10 5 (7.2) The mean square roughness (R s ) is the root of the average of the squares of deviations (y) from the mean surface. R s = 1 l l {y(x)}2 dx (7.3) 0