Module 1 : The equation of continuity. Lecture 4: Fourier s Law of Heat Conduction

Transcription

1 1 Module 1 : The equation of continuit Lecture 4: Fourier s Law of Heat Conduction NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering

2 Fourier s Law of Heat Conduction According to Fourier s law, the rate of heat flow, q, through a homogeneous solid is directl proportional to the area A, of the section at the right angles to the direction of the heat flow, and to the temperature difference T along the path of heat flow. Mathematicall, it can be written as q = k T (.17) Fourier s law of heat conduction is an empirical relationship based on observation. Equation (.16) holds for isotropic media. k is the thermal conductivit. The figure shown below illustrates the Fourier law of heat conduction. Fig..3 Heat flow through a homogeneous (isotropic) solid Heat Conduction Heat transfer b conduction involves transfer of energ within a bulk material without an motion of the material as a whole. Conduction takes place when a temperature gradient eist in a NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering

3 3 solid (or stationar fluid) medium. Energ is transferred from the more energetic to the less energetic molecules when neighboring molecules collide. Thermal conductivit Thermal conductivit (k) is the intrinsic propert of a material which relates its abilit to conduct heat. Thermal conductivit is defined as the quantit of heat (Q) transmitted through a unit thickness () in a direction normal to a surface of unit area (A) due to a unit temperature gradient ( T) under stead state conditions and when the heat transfer is dependant onl on temperature gradient. So mathematicall it can be written as k = q d (W/m K) (.18) dt where q = Rate of heat flow For non-isotropic media, q = k T, where k is the thermal conductivit tensor. Viscous Dissipation Function A function used to take into account the effect of the forces of viscous friction on the motion of a mechanical sstem. The dissipation function describes the rate of decrease of mechanical energ of the sstem. It is used, commonl, to allow for the transition of energ of ordered motion to the energ of disordered motion (ultimatel to thermal energ). For Newtonian fluids, the, where Φ is the viscous dissipation function. term: ( τ : ) = μ Φ NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering

4 NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering 4 In rectangular coordinates we can write it as 3 Φ = (.19) nder most circumstances the term μ Φ can be neglected. It ma be important in high speed boundar laer flow, where velocit gradient are large. Special Forms of the energ equation General assumptions a) Neglect viscous dissipation b) Assume constant thermal conductivit k Ideal gas An ideal gas is defined as one in which all collisions of atoms or molecules are perfectl elastic and in which there are no intermolecular attractive forces. In such a gas, all the internal energ is in the form of kinetic energ and an change in the internal energ is accompanied b a change in temperature.

5 5 An ideal gas can be characteried b three state variables: absolute pressure (P), volume (V) and absolute temperature (T). The relationship between them ma be deduce from the kinetic theor of gases and is called the Ideal Gas Law, which is given as PV = nrt (.0) where n is the number of moles of a gas and R is the niversal Gas Constant which is equal to J/mol K. I. Fluid at constant volume For ideal gas, Ρ T V = Ρ T (.1) Inserting equation (1.) into equation (.16) and neglecting viscous effects, then for ideal gas we get DT ρ Cv = k T-Ρ( ) Q (.) Dt II. Fluid at Constant Pressure At constant pressure, dp = 0 and maintaining all other conditions same as the previous case (I) we get NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering

6 6 DT ρ C p = k T Q (.3) Dt III. Fluid with densit independent of temperature According to Continuit equation, we can write ( ) = 0, and maintaining all other conditions same as the previous case (II) we get DT ρ C p = k T Q (.4) Dt IV. Solids The densit ma be considered constant and ( ) = 0.Then (1.7) ields T ρ C p = k T Q (.5) t Tpical Boundar Conditions on the Energ equation For heat transfer problems, there are five tpes of boundar condition at the solid surfaces that are pertinent. I. Initial Condition: At t=0 T (,,,0) = T (,, ) 0 Constant Surface Heat Flu: Here dependant variable (T) and its normal derivative are specified. k is the material thermal conductivit. NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering

7 7 Fig..4 heat transfer through composite wall It is also known as Neumann Condition. II. Here, Convection coefficient (h) is prescribed at the surface. In this B.C., dependant variable (T) and its normal derivative are connected b an algebraic equation. This tpe of B.C. is known as Robin Mied B.C. This is also an eample of flu tpe B.C. This tpe of B.Cs. are encountered in solid-fluid problems Fig..5 heat transfer at the fluid-solid interface III. Given temperature (ma be a function of time) or given flu at a surface. It can be given as NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering

8 8 T = 0 t (for stead state condition) IV. Here normal derivative of the dependant variable is specified at the plane surface or at the wall. Fig..6 heat transfer through insulated walls V. In this, normal derivative of the dependant variable is defined for clindrical or spherical geometr. Fig..7 heat transfer through walls of clinder or sphere NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering