One-Dimensional Heat Transfer - Unsteady

1 CM3110 Heat Transfer Lecture 311/6/20171 Faith A. Morrison, Michigan Tech Transport IPart II: Heat Transfer1 One-Dimensional Heat Transfer - UnsteadyProfessor Faith MorrisonDepartment of Chemical EngineeringMichigan Technological University Faith A. Morrison, Michigan Tech 1: Heat flux in a rectangular solid Temperature BCExample 2: Heat flux in a rectangular solid Newton s law of coolingExample 3: Heat flux in a cylindrical shell Temperature BCExample 4: Heat flux in a cylindrical shell Newton s law of coolingExample 5: Heat conduction with generationExample 6: Wall heating of laminar flowSUMMARYS teady State Heat TransferConclusion: When we can simplify geometry, assume steady state, assume symmetry, the solutions are easily obtainedCM3110 Heat Transfer Lecture 311/6/20172 Faith A.

2 Morrison, Michigan Tech 1: Heat flux in a rectangular solid Temperature BCExample 2: Heat flux in a rectangular solid Newton s law of coolingExample 3: Heat flux in a cylindrical shell Temperature BCExample 4: Heat flux in a cylindrical shell Newton s law of coolingExample 5: Heat conduction with generationExample 6: Wall heating of laminar flowSUMMARYS teady State Heat TransferConclusion: When we can simplify geometry, assume steady state, assume symmetry, the solutions are easily obtainedWhat about non steady state problems (quite common in heat Transfer )?Example 1:UnsteadyHeat Conduction in a Semi infinite solidA very long, very wide, very tall slab is initially at a temperature To.

3 At time t= 0, the left face of the slab is exposed to an environment at temperature T1. What is the time dependent temperature profile in the slab? The slab is a homogeneous material of thermal conductivity, k, density, , and heat capacity, Cp. Faith A. Morrison, Michigan Tech Heat Transfer : Unsteady StateCM3110 Heat Transfer Lecture 311/6/20173 Example 1:UnsteadyHeat Conduction in a Semi infinite solidA very long, very wide, very tall slab is initially at a temperature To. At time t= 0, the left face of the slab is exposed to an environmentat temperature T1. What is the time dependent temperature profile in the slab? The slab is a homogeneous material of thermal conductivity, k, density, , and heat capacity, Cp.

4 Faith A. Morrison, Michigan Tech is codefor: Newton s law of cooling boundary conditions: |flux| 1D Heat Transfer : Unsteady StateExample:UnsteadyHeat Conduction in a Semi infinite solidxyzHD Faith A. Morrison, Michigan Tech , D, very largeUnsteady State Heat Transfer6CM3110 Heat Transfer Lecture 311/6/20174xyoTTt 0oTTt 010 TTt 0(,)tTTxt xy Faith A. Morrison, Michigan Tech Condition:71D Heat Transfer : Unsteady StateGeneral Energy Transport Equation(microscopic energy balance)Vn dSSAs for the derivation of the microscopic momentum balance, the microscopic energy balance is derived on an arbitrary volume, V, enclosed by a surface, S. STkTvtTCp 2 Gibbs notation:see handout for component notation Faith A.

5 Morrison, Michigan Tech Heat Transfer : Unsteady StateCM3110 Heat Transfer Lecture 311/6/20175 General Energy Transport Equation(microscopic energy balance)see handout for component notationrate of changeconvectionconduction (all directions)sourcevelocity must satisfy equation of motion, equation of continuity(energy generated per unit volume per time)STkTvtTCp 2 Faith A. Morrison, Michigan Tech Heat Transfer : Unsteady StateNote: this handout is on the web: ~fmorriso/cm310 of energy for Newtonian fluids of constant density, , andthermal conductivity, k, with source term (source could be viscous dissipation, electricalenergy, chemical energy, etc., with units of energy/(volume time)).

6 CM310 Fall 1999 Faith MorrisonSource: R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Processes, Wiley, NY,1960, page notation (vector notation)ppCSTCkTvtT 2 Cartesian (xyz) coordinates:ppzyxCSzTyTxTCkzTvyTvxTvtT 222222 Cylindrical (r z) coordinates:ppzrCSzTTrrTrrrCkzTvTrvrTvtT 11 22222 Spherical (r ) coordinates:prrTrrTrrrCkTrvTrvrTvtTsin1s insin11 sin22222 Faith A. Morrison, Michigan Tech Heat Transfer Lecture 311/6/20176 Example 1:UnsteadyHeat Conduction in a Semi infinite solidA very long, very wide, very tall slab is initially at a temperature To. At time t= 0, the left face of the slab is exposed to an environment at temperature T1. What is the time dependent temperature profile in the slab?

7 The slab is a homogeneous material of thermal conductivity, k, density, , and heat capacity, Cp. Faith A. Morrison, Michigan Tech s law of cooling BC s:11 xyoTTt 0oTTt 010 TTt 0(,)tTTxt xy1D Heat Transfer : Unsteady StateppzyxCSzTyTxTCkzTvyTvxTvtT 222222 Microscopic Energy Equation in Cartesian Coordinates pCk thermal diffusivitywhat are the boundary conditions? initial conditions? Faith A. Morrison, Michigan Tech Heat Transfer : Unsteady StateCM3110 Heat Transfer Lecture 311/6/20177 Faith A. Morrison, Michigan Tech 7: UnsteadyHeat Conduction in a Semi infinite solidYou 0oTTt 010 TTt 0(,)tTTxt xyInitial Condition:1D Heat Transfer : Unsteady State 2222 xTxTCktTp Initial condition:Boundary conditions:xTTto ,0 0,01 tTThAqxxUnsteady State Heat Conduction in a Semi Infinite SlabtTTxo , Faith A.

8 Morrison, Michigan Tech U. Faith A. Morrison, Michigan Tech 0oTTt 010 TTt 0(,)tTTxt xy1D Heat Transfer : Unsteady StateCM3110 Heat Transfer Lecture 311/6/20178 2222 xTxTCktTp Initial condition:Boundary conditions:xTTto ,0 0,01 tTThAqxxtTTxo , Faith A. Morrison, Michigan Tech U. Faith A. Morrison, Michigan Tech 0oTTt 010 TTt 0(,)tTTxt xy for all Unsteady State Heat Conduction in a Semi Infinite Slab1D Heat Transfer : Unsteady State Faith A. Morrison, Michigan Tech U. Faith A. Morrison, Michigan Tech U. 2222 xTxTCktTp Initial condition:Boundary conditions:xTTto ,0 0,01 tTThAqxxtTTxo ,kth tx 2 16xyoTTt 0oTTt 010 TTt 0(,)tTTxt xyUnsteady State Heat Conduction in a Semi Infinite Slab1D Heat Transfer : Unsteady StateCM3110 Heat Transfer Lecture 311/6/20179 Geankoplis 4thed.

9 , eqn 7, page 363kth tx 2 complementary error function of error function of Faith A. Morrison, Michigan Tech erfc erfc erfc 1 erf erf 2 xyoTTt 0oTTt 010 TTt 0(,)tTTxt xySolution: Unsteady State Heat Conduction in a Semi Infinite Slab1D Heat Transfer : Unsteady StateGeankoplis 4thed., eqn 7, page 363kth tx 2 complementary error function of error function of Faith A. Morrison, Michigan Tech erfc erfc erfc 1 erf erf 2 xyoTTt 0oTTt 010 TTt 0(,)tTTxt xyTo make this solution easier to use, we can plot :1D Heat Transfer : Unsteady State Heat Conduction in a Semi Infinite SlabCM3110 Heat Transfer Lecture 311/6/201710 Faith A.

10 Morrison, Michigan Tech :Versus this:At various values of this:kth tx 2 erfc erfc xyoTTt 0oTTt 010 TTt 0(,)tTTxt xyTo make this solution easier to use, we can plot Heat Transfer : Unsteady State Heat Conduction in a Semi Infinite Slab tx 2 ooTTTT1 Geankoplis 4thed., Figure 3, page 364 Faith A. Morrison, Michigan Tech Heat Transfer : Unsteady State Heat Conduction in a Semi Infinite SlabCM3110 Heat Transfer Lecture 311/6 (), (), (), (),x1Y(),x5Y(),x10Y(),x50Y(),x100xx ooTTTT1increasing time, tt= , , , 1, 5, 10, 50, 100h= k= = 1 Unsteady State Heat Conduction in a Semi Infinite Slab Faith A. Morrison, Michigan Tech modern tools, we can plot the solution directly (evaluated on Mathcad)1D Heat Transfer : Unsteady State Heat Conduction in a Semi Infinite SlabHow could we use this solution?