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. 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 )?

2 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? 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.

3 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. 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.

4 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)).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.

5 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 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.

6 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., eqn 7, page 363kth tx 2 complementary error function of error function of Faith A.

7 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. 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.

8 , 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?Example:Will my pipes freeze?The temperature has been 35oF for a while now, sufficient to chill the ground to this temperature for many tens of feet below the surface. Suddenly the temperature drops to 20oF. How long will it take for freezing temperatures (32oF) to reach my pipes, which are 8 ftunder ground?

9 Use the following physical Faith A. Morrison, Michigan Tech heat Transfer : Unsteady State heat Conduction in a Semi Infinite SlabCM3110 heat Transfer Lecture 311/6/201712 Faith A. Morrison, Michigan Tech tx 2 erfc erfc xyoTTt 0oTTt 010 TTt 0(,)tTTxt xy ? ? ?? Both and depend on time1D heat Transfer : Unsteady State heat Conduction in a Semi Infinite Slabtx 2 ooTTTT1 Geankoplis 4thed., Figure 3, page 364 Faith A. Morrison, Michigan Tech (Interativesolution)1D heat Transfer : Unsteady State heat Conduction in a Semi Infinite SlabCM3110 heat Transfer Lecture 311/6/201713 Faith A. Morrison, Michigan Tech erfc erfc xyoTTt 0oTTt 010 TTt 0(,)tTTxt xyAnswer: 20 480 1D heat Transfer : Unsteady State heat Conduction in a Semi Infinite Slab2 HExample 8:UnsteadyHeat Conduction in a Finite sized solidxyzLD The slab is tall and wide, but of thickness 2H Initially at To at timet= 0 the temperature of the sides is changed to T1xy Faith A.

10 Morrison, Michigan Tech >0261D heat Transfer : Unsteady StateCM3110 heat Transfer Lecture 311/6/201714 Use same microscopic energy balance eqn as handout for component notationrate of changeconvectionconduction (all directions)source(energy generated per unit volume per time)STkTvtTCp 2 Unsteady State heat Transfer Faith A. Morrison, Michigan Tech heat Transfer : Unsteady heat Conduction in a Finite SolidppzyxCSzTyTxTCkzTvyTvxTvtT 222222 Microscopic Energy Equation in Cartesian Coordinates pCk thermal diffusivitywhat are the boundary conditions? initial conditions? Faith A. Morrison, Michigan Tech heat Transfer : Unsteady heat Conduction in a Finite SolidCM3110 heat Transfer Lecture 311/6/201715 Faith A. Morrison, Michigan Tech 8:UnsteadyHeat Conduction in a Finite sized solid1D heat Transfer : Unsteady heat Conduction in a Finite Solid 2222 xTxTCktTp Initial condition:Boundary conditions:xTTto ,00,2,011 tTTHxTTxUnsteady State heat Conduction in a Finite Slab Faith A.