Fourier's Law and the Heat Equation PDF

Fourier’s Law and the Heat Equation Chapter Two Fa2024 Dr. Islam Associate Professor/ME The Conduction Rate equation T1 T2 dT T T qx  k  k 2 1...

Fourier’s Law and the Heat Equation Chapter Two Fa2024 Dr. Islam Associate Professor/ME The Conduction Rate equation T1 T2 dT T T qx  k  k 2 1 dx L Recognizing that the heat flux is a vector quantity, we can write a more general statement of the conduction rate equation( Fourier's Law) as follows: Fourier’s Law Fourier’s Law A rate equation that allows determination of the conduction heat flux from knowledge of the temperature distribution in a medium Its most general (vector) form for multidimensional conduction is: q  k T Implications: – Heat transfer is in the direction of decreasing temperature (basis for minus sign). – Fourier’s law serves to define the thermal conductivity of the medium  k  q/ T      – Direction of heat transfer is perpendicular to lines of constant temperature (isotherms). – Heat flux vector may be resolved into orthogonal components. Heat Flux Components Cartesian Coordinates: T  x, y , z  T T T q  k i k j k k (2.3) x y z qx qy qz Cylindrical Coordinates: T r,  , z  T T T q  k i k j k k (2.24) r r z qr q qz Spherical Coordinates: T r,  ,  T T T q  k i k j k k (2.27) r r r sin   qr q q Heat Flux Components (cont.) In angular coordinates or  ,  , the temperature gradient is still based on temperature change over a length scale and hence has units of C/m and not C/deg. Heat rate for one-dimensional, radial conduction in a cylinder or sphere: – Cylinder qr  Ar qr 2 rLqr [W] or, qr  Arqr 2 rqr [W/m] – Sphere qr  Ar qr 4 r 2 qr [W] Heat Equation The Heat Diffusion Equation A differential equation whose solution provides the temperature distribution in a stationary medium. Based on applying conservation of energy to a differential control volume through which energy transfer is exclusively by conduction. Cartesian Coordinates:   T    T    T  T (2.19)  k    k    k   q  c p x  x  y  y  z  z  t Net transfer of thermal energy into the Change in thermal Thermal energy control volume (inflow-outflow) energy storage generation    T    T    T  T   k    k    k   q  c p x  x  y  y  z  z  t Heat Diffusion equation or heat equation, basic tool for heat conduction analysis Heat Equation (Special Case) One-Dimensional Conduction in a Planar Medium with Constant Properties and No Generation   T  T k   c p x  x  t becomes 2T 1 T  x 2  t k  thermal diffusivity of the medium  m 2 /s  cp   Heat Equation (Radial Systems) Cylindrical Coordinates: 1   T  1   T    T   T  kr  2  k    z  k z   q   c p (2.26) r r  r  r      t Spherical Coordinates: 1   2 T  1   T  1   T   T  kr    k    k sin    q   c p (2.29) r 2 r  r  r 2 sin 2      r 2 sin      t Boundary Conditions Boundary and Initial Conditions For transient conduction, heat equation is first order in time, requiring specification of an initial temperature distribution: T  x,t t=0 = T  x,0  Since heat equation is second order in space, two boundary conditions must be specified for each coordinate direction. Some common cases: Constant Surface Temperature: T 0 ,t  = Ts Constant Heat Flux: Applied Flux Insulated Surface T T -k |x=0= qs |x=0= 0 x x Convection: T -k |x=0= h  T - T 0,t  x Properties Thermophysical Properties Thermal Conductivity: A measure of a material’s ability to transfer thermal energy by conduction. Thermal Diffusivity: A measure of a material’s ability to respond to changes in its thermal environment. Property Tables: Solids: Tables A.1 – A.3 Gases: Table A.4 Liquids: Tables A.5 – A.7 Properties (Nanoscale Effects) Nanoscale Effects Conduction may be viewed as a consequence of energy carrier (electron or phonon) motion. For the solid state: 1 k  Cc mfp (2.7) 3 energy carrier mean free path → average distance specific heat per traveled by an energy carrier before unit volume. a collision. average energy carrier velocity, c < . Energy carriers also collide with physical boundaries, affecting their propagation.  External boundaries of a film of material. thick film (left) and thin film (right). Conduction Analysis Typical Methodology of a Conduction Analysis Consider possible microscale or nanoscale effects in problems involving small physical dimensions or rapid changes in heat or cooling rates. Solve appropriate form of heat equation to obtain the temperature distribution. Knowing the temperature distribution, apply Fourier’s law to obtain the heat flux at any time, location and direction of interest. Applications: Chapter 3: One-Dimensional, Steady-State Conduction Chapter 4: Two-Dimensional, Steady-State Conduction Chapter 5: Transient Conduction Example 2.3: The temperature distribution across a wall 1 m thick at a certain instant of time is given by: T  x  a  bx  cx 2 10/02/2024 16