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Overall Heat Transfer Coefficient- LMTD and NTU Methods- Fouling in Heat Exchangers- Heat Exchangers with Phase Change

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Overall Heat Transfer Coefficient- LMTD and NTU Methods- Fouling in Heat Exchangers- Heat Exchangers with Phase Change MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (Autonomous Institution – UGC, Govt. of India) Sponsored by CMR Educational Society (Affiliated to JNTU, Hyderabad, Approved by AICTE - Accredited by NBA & NAAC – A Grade - ISO 9001:2015 Certified) Maisammaguda, Dhulapally (Post Via. Kompally), Secunderabad – 500100, Telangana State, India. DEPARTMENT OF MECHANICAL ENGINEERING HEAT TRANSFER DIGITAL NOTES for B.TECH - III YEAR – II SEMESTER (2017-18) MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY L T/P/D C III Year B. Tech, ME-II Sem 5 1 4 (R15A0323) HEAT TRANSFER *Note: Heat and Mass Transfer data books are permitted Objectives: The objective of this subject is to provide knowledge about Heat transfer through conduction, convection and radiation. Student able to learn different modes of Heat Transfer. Student able to learn about the dimensional analysis . UNIT-I Introduction: Basic modes of heat transfer- Rate equations- Generalized heat conduction equation in Cartesian, Cylindrical and Spherical coordinate systems. Steady state heat conduction solution for plain and composite slabs, cylinders and spheres- Critical thickness of insulation- Heat conduction through fins of uniform and variable cross section- Fin effectiveness and efficiency. Unsteady state Heat Transfer conduction- Transient heat conduction- Lumped system analysis, and use of Heisler charts. UNIT-II Convection: Continuity, momentum and energy equations- Dimensional analysis- Boundary layer theory concepts- Free, and Forced convection- Approximate solution of the boundary layer equations- Laminar and turbulent heat transfer correlation- Momentum equation and velocity profiles in turbulent boundary layers- Application of dimensional analysis to free and forced convection problems- Empirical correlation. UNIT-III Radiation: Black body radiation- radiation field, Kirchhoff’s laws- shape factor- Stefan Boltzman equation- Heat radiation through absorbing media- Radiant heat exchange, parallel and perpendicular surfaces- Radiation shields. UNIT-IV Heat Exchangers: Types of heat exchangers- Parallel flow- Counter flow- Cross flow heat exchangers- Overall heat transfer coefficient- LMTD and NTU methods- Fouling in heat exchangers- Heat exchangers with phase change. Boiling and Condensation: Different regimes of boiling- Nucleate, Transition and Film boiling. Condensation: Laminar film condensation- Nusselt's theory- Condensation on vertical flat plate and horizontal tubes- Drop wise condensation. UNIT-V Mass Transfer: Conservation laws and constitutive equations- Isothermal equimass, Equimolal diffusion- Fick's law of diffusion- diffusion of gases, Liquids- Mass transfer coefficient. TEXT BOOKS: 1. Heat Transfer, by J.P.Holman, Int.Student edition, McGraw Hill Book Company. 2. Fundamentals of Heat and Mass Transfer- Sachdeva. 3. Heat transfer by Arora and Domakundwar, Dhanpat Rai & sons, New Delhi.. REFERENCE BOOKS: 1. Heat Transfer by Sukhatme. 2. Heat and Mass Transfer by R.K.Rajput, Laxmi Publications, New Delhi. 3. Heat transfer by Yunus A Cengel. OUTCOMES: Knowledge and understanding how heat and energy is transferred between the elements of a system for different configurations. Solve problems involving one or more modes of heat transfer. Student gets the exposure of different modes of Heat Transfer. UNIT-I Modes of Heat Transfer Heat Transfer by Conduction Fourier's Law of Heat Conduction dt Q =- Ka dx dt The temperature gradient is always negative along positive x direction and, dx therefore, the value as Q becomes + ve. Essential Features of Fourier’s law: 1. It is applicable to all matter (may be solid, liquid or gas). 2. It is a vector expression indicating that heat flow rate is in the direction of decreasing temperature and is normal to an isotherm. 3. It is based on experimental evidence and cannot be derived from first principle. Thermal Conductivity of Materials Sl. NO. Materials Thermal conductivity, (k) 1 Silver 10 W/mk 2 Copper 85 W/mk 3 Aluminium 25 W/mk 4 Steel 40 W/mk 5 Saw dust 0.07 W/mk 6 Glass wool 0.03 W/mk 7 Freon 0.0083 W/mk A. Pure metals, (k) = 10 to 400 W/mk Solid: B. Alloys, (k) = 10 to 120 W/mk C. Insulator, (k) = 0.023 to 2.9 W/mk Liquid: k = 0.2 to 0.5 W/mk Gas: k = 0.006 to 0.5 W/mk Thermal conductivity and temperature: k = k 0 (1+ βt ) Al ,U ( i ) Metals, k if t except. i.e. β ,−ve ( ii ) Liquid k if t except. H O 2 ( iii )Gas k if t ( iv ) Non -metal and i.e. ,+ ve β insulating material k if t various parameters on the thermal conductivity of solids. The following are the effects of various parameters on the thermal conductivity of solids. 1. Chemical composition: Pure metals have very high thermal conductivity. Impurities or alloying elements reduce the thermal conductivity considerably [Thermal conductivity of pure copper is 385 W/mºC, and that for pure nickel is 93 W/mºC. But monel metal (an alloy of 30% Ni and 70% Cu) has k of 24 W/mºC. Again for copper containing traces of Arsenic the value of k is reduced to 142 W/mºC]. 2. Mechanical forming: Forging, drawing and bending or heat treatment of metals causes considerable variation in thermal conductivity. For example, the thermal conductivity of hardened steel is lower than that of annealed state. 3. Temperature rise: The value of k for most metals decreases with temperature rise since at elevated temperatures the thermal vibrations of the lattice become higher that retard the motion of free electrons. 4. Non-metallic solids: Non-metallic solids have k much lower than that for metals. For many of the building materials (concrete, stone, brick, glass wool, cork etc.) the thermal conductivity may vary from sample to sample due Fire brick to variations in structure, composition, density and porosity. 5. Presence of air: The thermal conductivity is reduced due to the presence of air filled pores or cavities. 6. Dampness: Thermal conductivity of a damp material is considerably higher than that of dry material. 7. Density: Thermal conductivity of insulating powder, asbestos etc. increases with density Growth. Thermal conductivity of snow is also proportional to its density. Thermal Conductivity of Liquids V s k = 3 2 R Where = Boltzmann constant per molecule A v (Don’t confused with Stefen Boltzmann Constant) Vs = Sonic velocity of molecule = Distance between two adjacent molecule. R = Universal gas constant Av = Avogadro’s number Thermal conductivity of gas 1 k = nv f s 6 Where n = Number of molecule/unit volume v s = Arithmetic mean velocity f = Number of DOF = Molecular mean free path For liquid thermal conductivity lies in the range of 0.08 to 0.6 W/m-k For gases thermal conductivity lies in the range of 0.005 to 0.05 W/m-k The conductivity of the fluid related to dynamic viscosity () 4.5 k = 1 + 2n Cv ; l where, n = number of atoms in a molecule Sequence of thermal conductivity Pure metals > alloy > non-metallic crystal and amorphous > liquid > gases Wiedemann and Franz Law (based on experimental results) The ratio of the thermal and electrical conductivities is the same for all metals at the same temperature; and that the ratio is directly proportional to the absolute temperature of the metal. ’’ k k Where T or C α = T k = Thermal conductivity at T(K) υ υ υ = Electrical conductivity at T(K) − C = Lorenz number = 2.45 × 10 8 wΩ / k2 This law conveys that: the metals which are good conductors of electricity are also good conductors of heat. Except mica. Thermal Resistance: (Rth) Ohm’s Law: Flow of Electricity Voltage Drop = Current flow × Resistance Thermal Analogy to Ohm’s Law: T = qRth Temperature Drop = Heat Flow × Resistance A. Conduction Thermal Resistance: L (i) Slab ( R ) = th k A n ( r / r ) 2 1 (ii) Hollow cylinder ( R ) = th 2 kL π r r (iii) Hollow sphere ( R ) = 2 − 1 th 4 kr r π 1 2 1 B. Convective Thermal Resistance: ( R ) = th hA 1 C. Radiation Thermal Resistance: R = ( th ) 2 2 T + T2 )(T1 + T2 ) F A( 1 1D Heat Conduction through a Plane Wall (Thermal resistance) 1 L 1 R = t + + ∑ h1 A kA h2 A 1D Conduction (Radial conduction in a composite cylinder) 1D Conduction in Sphere Inside Solid: 1 d dT 2 kr 2 = 0 r dr dr 1 ( r1 / r) − T − T (r ) Ts , Ts ,1 s,2 = 1 − { } r1 / r2 1 − ( ) 4π k T − T qr = − kA dT = ( ( s ,1 s,)2 ) dr 1 / r1 − 1 / r2 = 1 / r1 1 / r2 R − t , cond 4 k π Isotropic & Anisotropic material If the directional characteristics of a material are equal /same, it is called an ‘Isotropic material’ and if unequal/different ‘Anisotropic material’. Example: Which of the following is anisotropic, i.e. exhibits change in thermal conductivity due to directional preferences? (a) Wood (b) Glass wool (c) Concrete (d) Masonry brick Answer. (a) ( ( ) Thermalconductivity k) Thermal diffusivity α = Thermalcapacity ( ρ c) 2 i. e . α = k unit m s c ρ The larger the value of α, the faster will be the heat diffuse through the material and its temperature will change with time. – Thermal diffusivity is an important characteristic quantity for unsteady condition situation. One Dimensional Steady State Conduction General Heat Conduction Equation in Cartesian Coordinates Recognize that heat transfer involves an energy transfer across a system boundary. A logical place to begin studying such process is from Conservation of Energy (1st Law of – Thermodynamics) for a closed system: dE i i Q W out = in − dt system The sign convention on work is such that negative work out is positive work in: i i dE = Q in + W out dt system The work in term could describe an electric current flow across the system boundary and through a resistance inside the system. Alternatively it could describe a shaft turning across the system boundary and overcoming friction within the system.
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