UNITS FOR THERMOMECHANICS

Transcription

1 UNITS FOR THERMOMECHANICS 1. Conitent Unit. Every calculation require a conitent et of unit. Hitorically, one et of unit wa ued for mechanic and an apparently unrelated et of unit wa ued for heat. For coupled thermomechanical analyi, one conitent et of unit covering both phenomena i neceary. 1.1 Mechanic. The deformed body occupie a region and i ubjected to urface force p per unit area and a gravity force g per unit ma and be in motion with particle acceleration a. The balance of linear momentum i expreed by pda + g d a d S =. (1.1) The relevant unit are Length (L), force (F), ma (M), and time (t). The reultant force acting on the urface i reultant applied force = pda. (1.) Therefore, the unit of tre i [ p] = FL. (1.) The total ma of the body with denity i Let M be the unit of ma, then, the unit of denity i The force of gravity i S total ma = d. (1.4) ML. = (1.) Therefore, The inertial force i Therefore, From (1.7) and (1.9), From (1.7) and (1.10), total gravity force = g d. (1.6) F = [g] M (1.7) reultant inertial force = a d. (1.8) F = MLt (1.9) g Lt = (1.10) F = MLt. (1.11) 1

2 That i, the force and math unit are not independent. Given a length unit and a time unit, we can chooe either the force unit or the ma unit but not both. 1. Thermodynamic The relevant unit are Length (L), force (F), ma (M), and time (t), and temperature (T). The total internal energy of the body i where the internal energy per unit ma i u: Uing (1.), The mechanical power of the urface force i U = u d (.1) U = u M. (.) Uing (1.), The balance of energy i expreed by P = p v da. (.) S 1 P F Lt =. (.4) v a d + p v g v n q u d = da d da r d + + S S (.) where the heat flux through an area i q; and the radiation energy aborbed per unit ma i r. Uing (1.) and (1.), Uing (.) and (1.11), For Fourier heat conduction, The unit of thermal conductivity i [ u] = L t 1 1 [ q] = [] r = L t U FL. (.6) = (.7) q = (.8) 1 1 [ ] = FT t (.9) The pecific heat per unit ma at contant train i u c = (.10) Therefore, the unit of pecific heat i 1 [] c =. (.11)

3 . SI Unit One conitent et of unit i the International Sytem of Unit called the SI ytem (Sytème International d'unité): The baic SI unit are Quantity Symbol SI unit Length L m (meter) Ma M kg (kilogram) Time t (econd) Temperature T K (degree Kelvin) Everything i to be expreed in thee baic unit. Force i a derived unit called the Newton (N). From (1.11), N = kg m (.1) The unit of tre i the Pacal (Pa). From (1.), Pa = N m (.) The unit of work and energy i the Joule (J). From (.7), J = N m (.) The unit of power i the Watt (W). From (.4) W = J 1 (.4) In ummary, the conitent unit are a follow. Quantity General Symbol SI unit Stre p FL Pa = N / m Denity 4 kg / m Gravity g Lt m/ Internal energy u Lt J / kg Heat flux q 1 1 W / m Radiation upply r Lt m / Conductivity 1 1 FT t N / ( K ) Specific heat c 1 m / ( K) Thermal expanion 1 T 1/ K

4 4. Engineering Unit Length and force are the mot often ued fundamental entitie in olid mechanic. In that cae, the fundamental unit are length, force, time, and temperature. Ma i then a derived unit. 4.1 Inch-Pound Sytem In engineering application in the USA, a common et of baic unit the following one: Quantity Symbol Inch-pound unit Length L in (inch) Force F lbf (pound force) Time t (econd) Temperature T R (degree Rankin) Everything i to be expreed in thee baic unit. Ma i a derived unit. The conitent unit are a follow. Quantity General Symbol Inch-pound ytem Stre FL lbf / in Denity 4 lbf / in 4 Gravity g Lt in / Internal energy u Lt in / Heat flux q 1 1 lbf / ( in ) Radiation upply r Lt in / Entropy 1 in / ( R) Conductivity 1 1 FT t lbf / ( R ) Specific heat c 1 in / ( R) Thermal expanion 1 T 1/ R Note that the thermal unit are not thoe commonly ued in heat tranfer. However, a conitent et of unit uch a thi one mut be ued for coupled thermo-mechanic calculation. ANSYS, for example, require the pecific heat at contant preure c p a input and then calculate the pecific heat at contant train c uing the other input propertie. An inconitent et of unit will caue an error in the calculation of c. 4

5 4. Millimeter-Newton Sytem The meter i too large a unit for machine deign and the Pacal i too mall a unit of tre. The millimeter i a more relevant length cale. In thi ytem, Called mega Pacal. The baic unit are a follow. p = N mm = N m = Pa = MPa (4.1) Quantity Symbol Millimeter-Newton Sytem Length L mm (millimeter) Force F N (Newton) Time t (econd) Temperature T K (degree Kelvin) Everything i to be expreed in thee baic unit. Ma i a derived unit. The conitent unit are a follow. Quantity General Symbol Millimeter-Newton Sytem Stre FL Denity 4 Gravity g MPa 4 N Lt mm / / mm Conductivity 1 1 FT t N / ( K ) Specific heat c 1 mm / ( K) Thermal expanion 1 T 1/ K. Converion Factor to the SI ytem The tandard meter, kilogram, and econd are fixed by international agreement. The tandard gravity i m g = The tandard inch i 1 in =. 04 m The ma of a ample weighing one pound i kg. From (1.7), the pound force i kg m 1 lbf = = N. The international BTU in term of Joule i 1 BTU = J

6 Horepower (HP) i here taken to be ` 1HP = 0 lbf ft lbf in = 6600 Thi reult in the following approximate converion factor. 1 in =. 04 m 1 m = in 1 lbf = N 1 N = lbf 1 ki = MPa 1 MPa = ki 1 lbm = kg 1 kg =. 0 lbm 9 1 R= K 1 K = R 9 1 lbf in = J 1 J = lbf in 4 1 BTU = 10 J 1 J = BTU 1 HP = W 1 W = HP 1 BTU = 98 lbf in 1HP = 669 lbf in / F 6. Structural Steel Typical propertie for teel in the preent ytem of unit are a follow. Property Inch-Pound ytem Millimeter-Newton ytem Elatic modulu, E lbf / in 10 Mpa Denity, lbf / in N / ( mm) c Specific heat, p in / ( R) ( mm) / ( K) Thermal conductivity, 7. 6 lbf / ( R) 60. N / ( K) Thermal expanion coef., R K Thi give the wave peed in teel: E in 10 =. 6