The Equation of Continuity and the Equation of Motion in Cartesian, cylindrical, and spherical coordinates

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1 The Equation of Continuit and the Equation of Motion in Catesian, clindical, and spheical coodinates CM30 Fall 20 Faith A. Moison Continuit Equation, Catesian coodinates t + v +v +v z v + + v + v z = 0 Continuit Equation, clindical coodinates Continuit Equation, spheical coodinates t + v + v θ + v z θ = 0 t + 2 v 2 + v θ sinθ + v φ sinθ θ sinθ φ = 0 Equation of Motion fo an incompessible fluid, 3 components in Catesian coodinates v t +v v +v v +v v z = P + τ + τ + τ z +g v t +v v +v v +v v z = P + τ + τ + τ z +g vz t +v v z +v v z +v v z z = P + τz + τ z + τ zz +g z Equation of Motion fo an incompessible fluid, 3 components in clindical coodinates v t +v v + v θ v θ v2 θ +v z vθ t +v v θ + v θ v θ θ + v θv v +v z v θ vz t +v v z + v θ v z θ +v z v z = P + τ + = P θ + 2 τ θ 2 + τ θθ = P + τ z + τ θz θ + τ zz τ θ θ τ θθ + τ z +g Equation of Motion fo an incompessible fluid, 3 components in spheical coodinates = = P + 2 τ 2 P θ + 3 τ θ 3 v t +v v + v θ v θ + v φ v sinθ φ v2 θ +v2 φ + τ θ sinθ + τ φ sinθ θ sinθ θ + τ zθ + τ θ τ θ +g θ +g z φ τ θθ + τ φφ +g vθ t +v v θ + v θ v θ θ + v φ v θ sinθ φ + v v θ v2 φ cotθ vφ + τ θθ sinθ + τ φθ sinθ θ sinθ φ + τ θ τ θ τ φφcotθ v φ + v θ v φ θ + v φ v φ sinθ φ + v v φ + v φv θ cotθ t +v = P sinθ φ + 3 τ φ 3 + τ θφ sinθ + τ φφ sinθ θ sinθ φ + τ φ τ φ + τ φθcotθ +g θ +g φ

2 Equation of Motion fo incompessible, Newtonian fluid Navie-Stokes equation 3 components in Catesian coodinates v t +v v +v v +v v z = P 2 +µ v v v 2 +g v t +v v +v v +v v z = P 2 +µ v v v 2 +g vz t +v v z +v v z +v v z z = P 2 +µ v z v z v z 2 +g z Equation of Motion fo incompessible, Newtonian fluid Navie-Stokes equation, 3 components in clindical coodinates v t +v v + v θ v θ v2 θ +v v z = P +µ v + 2 v 2 θ 2 2 v θ 2 θ + 2 v 2 +g vθ t +v v θ + v θ v θ θ + v v θ v θ +v z = P θ +µ v θ + 2 v θ 2 θ v 2 θ + 2 v θ 2 +g θ vz t +v v z + v θ v z θ +v v z z = P +µ v z + 2 v z 2 θ v z 2 +g z Equation of Motion fo incompessible, Newtonian fluid Navie-Stokes equation, 3 components in spheical coodinates v t +v v + v θ v θ + v φ v sinθ φ v2 θ +v2 φ = P +µ vθ t +v v θ + v θ v θ = P θ +µ vφ t +v v φ + v θ v φ = P sinθ φ +µ 2 2 v + 2 sinθ θ sinθ v θ 2 2 sinθ θ + v φ v θ sinθ φ + v v θ v2 φ cotθ 2 2v θ + 2 θ sinθ θ + v φ v φ sinθ φ + v v φ + v φv θ cotθ 2 2v φ + 2 θ sinθ + θ v θsinθ + 2 v 2 sin 2 θ φ 2 θ v θsinθ 2 2 sinθ 2 v θ 2 sin 2 θ φ v θ 2cotθ 2 sinθ θ v φsinθ + 2 v φ 2 sin 2 θ φ v 2 sinθ φ + 2cotθ 2 sinθ v φ +g φ v φ +g θ φ v θ φ +g φ Note: the -component of the Navie-Stokes equation in spheical coodinates ma be simplified b adding 0 = 2 v to the component shown above. This tem is zeo due to the continuit equation mass consevation. See Bid et. al. Refeences:. R. B. Bid, W. E. Stewat, and E. N. Lightfoot, Tanspot Phenomena, 2 nd edition, Wile: NY, R. B. Bid, R. C. Amstong, and O. Hassage, Dnamics of Polmeic Fluids: Volume Fluid Mechanics, Wile: NY, 987.

3 FACTORS FOR UNIT CONVERSIONS Pof. Faith A. Moison Depatment of Chemical Engineeing Quantit Mass Equivalent Values kg = 000 g = 0.00 metic ton = lb m = oz lb m = 6 oz = ton = g = kg Length m = 00 cm = 000 mm = 0 6 micons µm = 0 0 angstoms Å = in = ft =.0936 d = mile ft = 2 in. = /3 d = m = cm Volume m 3 = 000 lites = 0 6 cm 3 = 0 6 ml = ft 3 = impeial gallons = gal = qt ft 3 = 728 in 3 = gal = m 3 = lites = cm 3 Foce N = kg. m/s 2 = 0 5 dnes = 0 5 g. cm/s 2 = lb f lb f = lb m. ft/s 2 = N = dnes Pessue atm = N/m 2 Pa = kpa =.0325 bas = dnes/cm 2 = 760 mm Hg at 0 C to = m H 2 O at 4 C = lb f /in 2 psi = 33.9 ft H 2 O at 4 C 00 kpa = ba Eneg J = N. m = 0 7 egs = 0 7 dne. cm = kw. h = cal = ft. lb f = Btu Powe W = J/s = cal/s = ft. lb f /s = Btu/s = 3.42 Btu/h = hp hosepowe Viscosit Pa. s = N. s/m 2 = kg/m. s = 0 poise = 0 dnes. s/cm 2 = 0 g/cm. s = 0 3 cp centipoise = lb m /ft. s = lb m /ft. h Densit kg/m 3 = 0 3 g/cm 3 = lb m /ft kg/m 3 = g/cm 3 = lb m /ft 3 Volumetic Flow m 3 /s= ft 3 /s=5,850.2 gal/min gpm gpm = m 3 /s= ft 3 /s= lite/min lite/min= gpm Ve. 2 Sep 20

4 o 5 o Tempeatue T C 9 T F 32 T F T C 32.8 T C 32 o 9 o o 5 Absolute Tempeatue TK = T C T R = T F Tempeatue Inteval T C = K =.8 F =.8 R F = R = 5/9 C = 5/9 K USEFUL QUANTITIES SG = 20 C/ wate 4 C wate 4 C = 000 kg/m 3 = lb m /ft 3 =.000 g/cm 3 wate 25 C = kg/m 3 = lb m /ft 3 = g/cm 3 g = m/s 2 = cm/s 2 = ft/s 2 µ wate 25 C = Pa. s = kg/m. s = cp = g/cm. s = lb m /ft. s Composition of ai: N % O % A 0.94% CO % H 2, He, Ne, K, Xe 0.0% 00.00% M ai = 29 g/mol = 29 kg/kmol = 29 lb m /lbmole Ĉ p,wate 25 C = 4.82 kj/kg K = cal/g C = Btu/lb m F R = 8.34 m 3. Pa/mol. K = lite. ba/mol. K = lite. atm/mol. K = lite. mm Hg/mol. K = ft 3. atm/lbmole. R = 0.73 ft 3. psia/lbmole. R = 8.34 J/mol. K =.987 cal/mol. K =.987 Btu/lbmole. R Ve. 2 Sep 20

5 Data Coelations fo Eaminations CM30 Tanspot Phenomena I Michigan Technological Univesit Pofesso Faith A. Moison I. Flow though Smooth Pipes A. All Renolds numbes: Moison The coelation fom Moison 203 fits the smooth pipe data fo all Renolds numbes; beond 4000 this coelation follows the Pandtl equation see Figue ; Moison, equation This coelation is eplicit in ; when flow ate is known, Δ ma be found diectl; when Δ is known, o must be solved fo iteativel. Moison B., : Pandtl The Pandtl coelation fo in tubulent flow is not eplicit in fiction facto and must be solved iteativel ecept when is known Moison, equation This is good onl fo 4,000/ Figue : Equation 3 captues smooth pipe fiction facto as a function of Renolds numbe ove the entie Renolds numbe ange Moison, 203 and is ecommended fo speadsheet use. Also shown ae Nikuadse's epeimental data fo flow in smooth pipes Nikuadse, 933. Use beond 0 is not ecommended; fo Re>4000 equation 3 follows the Pandtl equation. Pandtl o VonKaman Nikuadse Denn, log C., : A simplified Coelation Fo the tubulent egime, an appoimate coelation that is much simple to wok with with a calculato on an eam, fo eample is given hee and shown in Figue 2 Moison, equation This is good onl fo 4,000. Simplified Tubulent White, log. 3 f Lamina flow 6 f Re Simplified Coelation: Tubulent flow Smooth pipe Nikuadse data Lamina flow Pandtl Re>4,000 Simplified Re>4,000 Pandtl Coelation 4.0log Re f 0.40 f 0.00.E E E E E E Re Figue 2: Fo tubulent flow, the simplified equation 3 o Pandtl equation 2 coelations ma be used. Fo wok with a calculato, the simplified coelation is pehaps the easiest to wok with. CM30 Moison Michigan Tech 203 CM30 Moison Michigan Tech 203 2

6 II. Flow Aound a Sphee A. All Renolds Numbes: Moison The coelation fom Moison 203 fits the flow aound a sphee fo all Renolds numbes Figue 3; Moison equation 8.83; beond 0 this coelation follows the cuve shown in Figue 3.. Moison , , Simplified Coelations The coelations below Moison, 203; equation 8.82 ae simple elationships moe suitable to calculato/eam wok. 2 0.,000, log log Figue 3: Equation 4 captues flow aound a sphee as a function of Renolds numbe ove the entie Renolds numbe ange Moison, 203 and is ecommended fo speadsheet use. Also shown ae epeimental data fom White 974. Use beond 0 is not ecommended. Refeences M. Denn, Pocess Fluid Mechanics Pentice Hall, Englewood Cliffs, NJ, 980 F. A. Moison, An Intoduction to Fluid Mechanics Cambidge Univesit Pess, New Yok, 203. F. M. White, Viscous Fluid Flow McGaw Hill, Inc.: New Yok, 974. CM30 Moison Michigan Tech CM30 Moison Michigan Tech 203 4

8 Fom F. A. Moison, An Intoduction to Fluid Mechanics Cambidge, NY 203

The Equation of Continuity and the Equation of Motion in Cartesian, cylindrical, and spherical coordinates

The Equation of Continuity and the Equation of Motion in Catesian, cylindical, and spheical coodinates CM3110 Fall 2011 Faith A. Moison Continuity Equation, Catesian coodinates t + v x x +v y y +v z vx