Units and Dimensions PDF

Units and Dimensions Jose V Parambil 1 CB2101 – Aug 2024 Units & Dimensions Units Basis for any calculation or measurement Metric system (France, 1795) CGS sy...

Units and Dimensions Jose V Parambil 1 CB2101 – Aug 2024 Units & Dimensions Units Basis for any calculation or measurement Metric system (France, 1795) CGS system (centimeter-gram-second) MTS system (metre-tonne-second) MKS system (metre-kilogram-second) MMGS system (millimeter-gram-second) English units (pre 1824) Imperial units (post 1824) United States customary system (1832) International System of Units (SI units) (1960) 7 base units Derived units (products of base units) J V Parambil, CBE IIT Patna 2 CB2101 – Aug 2024 Units & Dimensions Basis of units – Temperature Fahrenheit scale (Daniel Gabriel Fahrenheit, 1724) 212 °F – Boiling point of water 96 °F – Human body temperature 98.6 °F – Human body temperature 32 °F – Freezing point of water 0 °F – Mixture of ice, water, ammonium chloride Centigrade/Celsius scale (Anders Celsius, 1742) Absolute temperature scale (Lord Kelvin, 1848) J V Parambil, CBE IIT Patna 3 CB2101 – Aug 2024 Units & Dimensions Basis of units – Power Horsepower (James Watt, 1782) Rotates a mill wheel of 12 m 144 times per hour, using a force of 180 pounds-force. 𝑊 𝐹𝑑 180 𝑙𝑏𝑓 × 2𝜋 × 12𝑓𝑡 × 144 𝑓𝑡. 𝑙𝑏𝑓 𝑁. 𝑚 𝑃= = = = 32,555 ≈ 736 𝑡 𝑡 60 𝑚𝑖𝑛 𝑚𝑖𝑛 𝑠 1 𝑙𝑏𝑓 ≈ 4.448 𝑁; 1 𝑓𝑡 ≈ 0.305 𝑚 Current conversion: 1 Mechanical Horsepower = 745.7 W Watt (1882) 𝐽 𝑁. 𝑚 𝑘𝑔. 𝑚2 1𝑊 =1 =1 =1 1 𝑊 = 1 𝑉. 𝐴 𝑠 𝑠 𝑠3 J V Parambil, CBE IIT Patna 4 CB2101 – Aug 2024 Units & Dimensions SI Units – base units Quantity (Dimen. symbol) Unit (symbol) Defining constant (symbol) Constant Value Time (T) second (s) Transition frequency (𝚫𝝂𝑪𝒔 ) 9192631770 Hz Length (L) metre (m) Speed of Light (c) 299792458 m/s Mass (M) kilogram (kg) Plank constant (h) 6.62607015×10−34 J⋅s Electric current (I) ampere (A) Elementary charge (e) 1.602176634×10−19 C Therm. Temperature (𝚯) kelvin (K) Boltzmann constant (k) 1.380649×10−23 J/K Amount of Substance (N) mole (mol) Avogadro constant (NA) 6.02214076×1023 mol−1 Luminous efficacy of 540 THz Luminous Intensity (J) candela (cd) radiation (Kcd) 683 lm/W J V Parambil, CBE IIT Patna 5 CB2101 – Aug 2024 Units & Dimensions Redefinition of SI base units (2019) Prior to 2019 After 2019 (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units) J V Parambil, CBE IIT Patna 6 CB2101 – Aug 2024 Units & Dimensions Dimensions Generalizes a physical quantity Quantity Dimension Independent of unit Density Velocity Acceleration Force Work Power Dimensional Analysis A theoretical equation relating different physical quantities must be dimensionally homogeneous. Dimensionless Numbers J V Parambil, CBE IIT Patna 7 CB2101 – Aug 2024 Units & Dimensions Rayleigh Method of Dimensional Analysis The force (F) exerted on a body immersed in a flowing fluid is dependent on the relative velocity (u) between the solid and the fluid, the fluid density (ρ), the fluid viscosity (μ), and a characteristic length of the body (l). Determine the dimensionless groups in which the dimensional variables may be arranged. 𝐹 ∝ 𝑢𝑎 𝜌𝑏 𝜇𝑐 𝑙^𝑑 𝐹 = 𝐾𝑢𝑎 𝜌𝑏 𝜇𝑐 𝑙 𝑑 𝐹 = 𝑢 = 𝜌 = 𝜇 = 𝑙 = J V Parambil, CBE IIT Patna 8 CB2101 – Aug 2024 Units & Dimensions Dimensionless Numbers (https://www.3blmedia.com/news/new-gm-wind-tunnel-accelerates-fuel-saving-designs) J V Parambil, CBE IIT Patna 9 CB2101 – Aug 2024 Units & Dimensions Dimensionless Numbers Independent of units. Similarity in different scales. (https://www.imperial.ac.uk/aeronautics/research/facilities/10x5-wind-tunnel/projects/) J V Parambil, CBE IIT Patna 10 CB2101 – Aug 2024 Units & Dimensions Dimensionless Numbers in Chemical Engineering Geometric similarity Kinematic similarity Name Equation Significance Dynamic similarity 𝑢𝐷 Reynolds No. Inertial/viscous forces 𝜈 𝜈 Prandtl No. Momentum/thermal diffusivity Heat, Mass, and Momentum 𝛼 transfers, Reaction rates Nusselt No. ℎ𝐿 Convective/conductive heat transfer 𝑘 𝑘𝐺 𝐿 Friction factor Sherwood No. Total/diffusive mass transfer 𝐷𝐴𝐵 Fanning Thiele Modulus Reaction/diffusion rate Darcy-Weisbach Fanning Friction Δ𝑝𝐷 Pressure/inertial forces 𝑓𝐷 = 4𝑓 Factor 2𝐿𝑢2 𝜌 J V Parambil, CBE IIT Patna 11 CB2101 – Aug 2024 Units & Dimensions Scale Up How to achieve a similar quality and yield of product across the scales? Target to achieve a Maintain consistency in quality. similar degree of: Secure confidence in process. Mass transfer, Flow behavior, Mixing Geometric similarity. Reasonable cooling and heating Temperature rates, heat transfer area, mixing. Catalyst contact area, mixing, Reaction rate process conditions. J V Parambil, CBE IIT Patna 12 CB2101 – Aug 2024 Units & Dimensions Scale Up Mixing 𝑁𝐷𝑖2 𝜌 Turbulent when > 10,000. Reynolds Number 𝑁𝑅𝑒 = 𝜇 Ensures similar flow pattern and heat transfer coefficient. Np is function of NRe. Power Number 𝑃 = 𝑁𝑃 𝜌𝑁 3 𝐷𝑖5 Measures energy delivered to the fluid Important for dispersions. Flow Number 𝑄 = 𝑁𝑄 𝑁𝐷𝑖3 Measure of pumping action of impeller. Highest fluid velocity and shear. Tip Speed 𝑣𝑖,𝑡𝑖𝑝 = 𝜋𝐷𝑖 𝑁 Important for dispersions. Choose similar geometry, Fix scaleup criteria. Determine agitation rate. agitator type. J V Parambil, CBE IIT Patna 13