Presentation on theme: "Transient Conduction: Spatial Effects and the Role of Analytical Solutions Chapter 5 Sections 5.4 to 5.7 Lecture 10."— Presentation transcript:
1 Transient Conduction: Spatial Effects and the Role of Analytical Solutions Chapter 5Sections 5.4 to 5.7Lecture 10
2 Solution to the Heat Equation for a Plane Wall with Symmetrical Convection ConditionsIf the lumped capacitance approximation cannot be made, consideration mustbe given to spatial, as well as temporal, variations in temperature during thetransient process.For a plane wall with symmetrical convectionconditions and constant properties, the heatequation and initial/boundary conditions are:(5.29)(5.30)(5.31)(5.32)Existence of eight independent variables:(5.33)How may the functional dependence be simplified?
3 Dimensionless temperature difference: Plane Wall (cont.)Non-dimensionalization of Heat Equation and Initial/Boundary Conditions:Dimensionless temperature difference:Dimensionless space coordinate:Dimensionless time:The Biot Number:Exact Solution:(5.42a)(5.42b,c)See Appendix B.3 for first four roots (eigenvalues ) of Eq. (5.42c).
4 The One-Term Approximation : Plane Wall (cont.)The One-Term Approximation :Variation of midplane temperature (x*= 0) with time :(5.44)Variation of temperature with location (x*) and time :(5.43b)Change in thermal energy storage with time:(5.46a)(5.49)(5.47)Can the foregoing results be used for a plane wall that is well insulated on oneside and convectively heated or cooled on the other?Can the foregoing results be used if an isothermal condition isinstantaneously imposed on both surfaces of a plane wall or on one surface ofa wall whose other surface is well insulated?
5 Graphical Representation of the One-Term Approximation Heisler ChartsGraphical Representation of the One-Term ApproximationThe Heisler Charts, Section 5 S.1Midplane Temperature:
6 Temperature Distribution: Heisler Charts (cont.)Temperature Distribution:Change in Thermal Energy Storage:
7 Radial Systems Long Rods or Spheres Heated or Cooled by Convection. One-Term Approximations:Long Rod: Eqs. (5.52) and (5.54)Sphere: Eqs. (5.53) and (5.55)Graphical Representations:Long Rod: Figs. 5 S.4 – 5 S.6Sphere: Figs. 5 S.7 – 5 S.9
8 The Semi-Infinite Solid A solid that is initially of uniform temperature Ti and is assumed to extendto infinity from a surface at which thermal conditions are altered.Special Cases:Case 1: Change in Surface Temperature (Ts)(5.60)(5.61)
10 Example 1 (Problem 5.34)A 100-mm-thick steel plate (=7830 kg/m3, c=550 J/kgK,k=48 W/mK) that is initially at uniform temperature ofTi=200C is to be heated to a minimum temperature of 550C.Heating is affected in a gas-fired furnace, where the productsof combustion at T = 800 C maintain a convection coefficientof h=250 W/m2K on both surfaces of the plate, how long should theplate be left in the furnace ? What is the surface temperature?
11 Example 1 Known: Dimension and properties of steel plate, convection condition.Find: Time needed to reach Tmin of 550C, and T surfaceSchematic:
12 Example 1Assumptions: 1-D conduction in the plate, radiation negligibleConstant properties.Analysis:Lumped Capacitance Method (Need Bi<0.1)
13 Example 1 Analysis: 2. Approximate Solution (Need Fo>0.2) The minimum T is at the center of the plateFrom Table 5.1, Bi=0.26, ζ1=0.488 rad, C1=1.0396Solve Fo=3.839 >0.2
26 Problem: Thermal Response Firewall Problem: 5.93: Use of radiation heat transfer from high intensity lampsfor a prescribed duration (t=30 min) to assessability of firewall to meet safety standards corresponding tomaximum allowable temperatures at the heated (front) andunheated (back) surfaces.
28 Problem: Thermal Response of Firewall (cont.) <
29 Problem: Microwave Heating Problem: : Microwave heating of a spherical piece of frozen ground beef using microwave-absorbing packaging material.
30 Problem: Microwave Heating (cont.) ThusThe beef can be seen as the interior of a sphere with a constant heat flux at its surface, thusthe relationship in Table 5.2b, Interior Cases, sphere, can be used. We begin by calculating q* for Ts=0°C.
31 Problem: Microwave Heating (cont.) Since this is less than 0.2, our assumption was correct. Finally we can solvefor the time:<COMMENTS: At the minimum surface temperature of -20°C, with T∞ = 30°C and h = 15 W/m2∙Kfrom Problem 5.33, the convection heat flux is 750 W/m2, which is less than 8% of the microwaveheat flux. The radiation heat flux would likely be less, depending on the temperature of the oven walls.