Principles of Food and Bioprocess Engineering (FS 231) Example Problems on Units and Dimensions 1. Determine the dimensions of the following quantities starting from their units: a. Work b. Specific heat c. Enthalpy d. Power e. Density f. Thermal conductivity g. Pressure h. Viscosity i. Latent heat j. Overall heat transfer coefficient 2. Convert: a. 3 J/kg /C to J/kg K b. 50 in 2 to m 2 c. 20 psi to Pa d. 15 lb m /ft 3 to kg/m 3 e. 10 ft lb f /g /F to kj/kg K 3. Convert the following quantities to their corresponding SI units: a. 5 g/cm s b. 20 dynes/cm 2 c. 5 lb f /ft hr d. 5 Btu/hr ft /F e. 7 lb f /ft 2 /F f. 4 Btu/lb m mm Hg g. 8 cal/in 3 hr h. 6 hp/cm 2 /C i. 3 Btu/min in 2 j. 6 lb f min/ft 2 k. 7 cal/in 2 min l. 4 hp/mm Hg cm 2 /F m. 23 Btu lb m /lb f cm 2 hr 3 /F n. 14 Btu g/lb f in 2 hr 3 /F o. 2 ft 3 /gal lb f p. 1.4 mm Hg/in 2 g cm 2 /F q. 6.3 Btu/lb m /F r. 8.7 cal cm/psi s. 5 Btu psia cm/lb m /F hr t. 3 Btu ft min/lb f /F u. 5 Btu min/lb f in hr /C v. 5 J/g /F w. 3 Btu/lb m x. 2 Btu/ft min /F y. 6.7 hp cal/mm 3 lb f /F z. 2.8 dyne in 2 /g mm Hg aa. 0.45 psia/gal lb m /C bb. 23 g min/cal cm 4 Btu cc. 23.5 psia Btu/in 3 lb m /F dd. 0.8 g cal 2 /hp bar hr ee. 7.4 psia/dyne (mm Hg) ff. 3.5 cm min/lb f cc ft 4. Is the following equation dimensionally consistent? EXPLAIN your answer. H 1 = H 2 + (H 3 )*(H 4 ) where H denotes enthalpy. 5. Are the following equations dimensionally consistent? Start from the dimensions of individual quantities. a. Force = (Viscosity) x (Radius) x (Velocity) b. Energy = (Thermal conductivity) x (Area) x (Temperature) c. Pressure = [(Viscosity) x (Velocity)] + [Length] d. Power = (Density) x (Volume) x (Specific heat) x (Temperature) / (Time) e. Energy = [(Heat transfer coefficient) x (Area) x (Temperature)] / [(Heat transfer coefficient) x (Area) x (Temperature)] f. Enthalpy = (Force x Velocity) / (Viscosity x Length) g. Thermal diffusivity = Area + (frequency) 2 h. (Specific heat) (shear rate) (mass) = [(Torque) (acceleration) (time)] / [(area) (Temperature)] i. (Heat transfer coefficient)/(velocity) = [(Work)/(Volume)] + [1/ (Temperature)] j. Volumetric Flow Rate = (Power) / (Specific Heat x Change in Temperature) k. Pressure = (Thermal conductivity) x (Time) x (Temperature) x (Density) x (Length)/(mass) 6. Are the following equations dimensionally consistent? Start from the units and not from the dimensions. a. Work = [(Density x Area x Diffusion coefficient x Force) / (Mass x Frequency)] +
[(Latent heat x Mass flow rate x time)] b. Velocity = [Acceleration] + [(Mass flow rate x Length) / (Mass)] c. Heat transfer coefficient = (Torque x Shear rate) / (Area x Temperature) d. Permeability coefficient = [(Area)] + [(Length x Thermal diffusivity x Pressure) / (Volume)] e. Enthalpy = (Specific heat) (Temperature) + (Work done)/(mass) f.(density)(area) 3 + (Length) 4 (Thermal Conductivity)(Time) 2 (Temperature) = (Viscosity)(Velocity)(Volume) g. (Power)/(Specific Heat) = (Mass Flow Rate)(Temperature) + (Density)(Volume)/(Time) 7. The term friction factor (f) is defined by the following equation: where )P is the pressure drop D is the diameter D is the density L is the length u is the velocity a. What are the dimensions and units of 'f '? b. If )P = 200 psi D = 2 cm D = 20 lb m /gal L = 30 m u = 350 cm/s What is the value of 'f '? 8. Show that m 2 /s 2 and J/kg are identical units. 9. The expression for energy transferred from a hot fluid to a solid object is given by: Where is the mass flow rate of the fluid, c p is the specific heat, and )T is the difference in temperature between the fluid and the object. What are the units of Q? 10. Reynolds number, N Re, given by the following equation, is used to characterize flow in a tube: Where D is the density, is the average velocity, D is a characteristic length, and : is viscosity. What are the units of Reynolds number? 11. The following equation is used to determine the Grashof number (N Gr ), which is a dimensionless quantity, during free convection:
D (diameter) = 2 inches D (density) = 1.5 g/cm 3 g (acceleration due to gravity) = 32.2 ft/s 2 $ (coef. of vol. thermal exp.) = 3.5 x 10-3 K -1 )T (temperature difference) = 120 /F : (viscosity) = 0.025 cp Determine the Grashof number (N Gr ) in this problem. 12. The following equation is used to determine the condensation number (N Cd ), which is a dimensionless quantity used in condensation applications: g (acceleration due to gravity) = 32.2 ft/s 2 D (density) = 0.5 lb m /in 3 (Latent heat of vaporization) = 2.2 x 10 6 Btu/g L (Characteristic length) = 3.5 inches k (Thermal conductivity) = 0.0013 hp/ft /F : (Viscosity) = 7.4 x 10-8 lb f min/ft 2 )T (Difference in temperature) = 85 /F Determine the condensation number (N Cd ) in this problem. 13. The following equation is used to determine the Fourier number (N Cd ), which is a dimensionless quantity used in heat transfer applications: 14. k (Thermal conductivity) = 0.0016 Btu cm/in 2 min /F t (Time) = 0.35 h D (Density) = 200 lb m /gal c p (Specific heat) = 2.7 Btu/g /F D (Characteristic length) = 35 mm Determine the Fourier number (N Fo ) in this problem. If Force = 20 dynes Viscosity = 10 cp Energy = 35 Btu Specific Heat = 60 Btu/lb m /C T 1 = 50/F
T 2 = 60/F Volume = 45 cm 3 Power = 55 Btu/min Density = 2 g/cm 3 Velocity = 25 cm/s Time = 0.5 min Determine the value of Pressure (in SI units). 15. The following equation is used to determine the viscosity (:) of a fluid: T = 2000 dyne cm N = 60 rpm L = 35 cm R i = 5 cm R o = 7 cm Determine the viscosity (:) of the fluid (in SI units). 16. The following equation is used to determine the energy content within a system: (Mass flow rate) = 23.5 lb m /min c p (Specific heat) = 16.4 Btu/g /F T (Temperature) = 12 /F a. Determine Q (in SI units) in this problem. b. If T in the above equation is Temperature difference instead of Temperature, what would the value of Q (in SI units) be? 17. The following equation is used to determine the energy content within a system: h (Heat transfer coefficient) = 0.00235 Btu/in 2 /F A (Surface area) = 75.8 ft 2 )T (Temperature difference) = 32 /F Determine Q (in SI units).
Answers to Example Problems on Units and Dimensions 1. a. ML 2 T -2 b. L 2 T -2 K -1 c. L 2 T -2 d. ML 2 T -3 e. ML -3 f. MLT -3 K -1 g. ML -1 T -2 h. ML -1 T -1 i. L 2 T -2 j. MT -3 K -1 2. a. 3 J/kg K b. 0.0323 m 2 c. 137900 Pa d. 240.28 kg/m 3 e. 24.4 kj/kg K 3. a. 0.5 kg/m s b. 2 N/m 2 c. 0.02 N/m s d. 8.7 W/m K e. 603.3 N/m 2 K f. 69.78 J/kg Pa g. 567.4 J/m 3 s h. 44.74 x 10 6 W/m 2 K i. 81,762.7 W/m 2 j. 17,236.8 Pa s k. 756.6 W/m 2 l. 402,711 m/s K m. 9.55 x 10-4 J kg/n m 2 s 3 K = 9.55 x 10-4 kg/m s 3 K = 9.55 x 10-4 J/m 3 s K n. 1.99 x 10-7 J kg/n m 2 s 3 K = 1.99 x 10-7 kg/m s 3 K = 1.99 x 10-7 J/m 3 s K o. 3.36 N -1 p. 5.2 x 10 12 Pa/m 4 kg K q. 26375.6 J/kg K r. 5.3 x 10-5 J m/pa = 5.3 x 10-5 m 4 s. 400.93 J Pa m/kg K s t. 23422.2 J m s/n K u. 778.1 J/N m K v. 9000 J/kg K w. 6977.7 J/kg x. 207.7 W/m K y. 8.46 x 10 12 W J/m 3 N K z. 1.36 x 10-7 m 4 /kg aa. 1.81 x 10 6 Pa/m 3 Kg bb. 31264.4 kg s/j 2 m 4 cc. 4.14 x 10 13 Pa J /m 3 kg K dd. 5.2 x 10-14 kg J 2 /W Pa s ee. 3.83 x 10 7 N -1 ff. 1.55 x 10 6 s 3 /kg m 4 4. No 5. a. Yes b. No c. No d. Yes e. No f. Yes g. No h. No i. No j. No k. No 6. a. Yes b. No c. Yes d. No e. Yes f. No g. Yes 7. a. Dimensionless b. 0.0157 8. 9. J/s (or Watts) 10. No units 11. N Gr = 1.08 x 10 12 12. N Cd = 5.33 x 10 19 13. N Fo = 6.57 x 10-6 14. 1404.5 Pa 15. 0.0014 Pa s 16. a. 1.38 x 10 9 J/s b. 3.5 x 10 7 J/s 17. 866.8 kj