The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 3
Текст из файла (страница 3)
This ratio is denotedby the Biot number, Bi.Bi =Rk hc (V A)=Rck(4.1.26)The temperature (which will be uniform throughout the body at any time for this situation) responsewith time for this system is given by the following relationship. Note that the shape of the body is notimportant — only the ratio of its volume to its area matters.T (t ) - T¥= e - hc At rVcTo - T¥(4.1.27)Typically, this will hold for the Biot number being less than (about) 0.1.Bodies with Significant Internal ResistanceWhen a body is being heated or cooled transiently in a convective environment, but the internal thermalresistance of the body cannot be neglected, the analysis becomes more complicated.
Only simplegeometries (a symmetrical plane wall, a long cylinder, a composite of geometric intersections of thesegeometries, or a sphere) with an imposed step change in ambient temperature are addressed here.The first geometry considered is a large slab of minor dimension 2L. If the temperature is initiallyuniform at To, and at time 0+ it begins convecting through a heat transfer coefficient to a fluid at T¥, thetemperature response is given by¥q=2æösin l n Lå çè l L + sin l L cos l L ÷ø exp(-l L Fo) cos(l x)n =1nn2 2nn(4.1.28)nand the ln are the roots of the transcendental equation: lnL tan lnL = Bi.
The following definitions hold:© 1999 by CRC Press LLC4-10Section 4Bi ºhc LkFo ºatL2qºT - T¥To - T¥The second geometry considered is a very long cylinder of diameter 2R. The temperature responsefor this situation is¥q = 2Bi(å (l Rn =1)exp - l2n R 2 Fo J o (l n r )2n2(4.1.29))+ Bi J o (l n R)2Now the ln are the roots of lnR J1(lnR) + Bi Jo (lnR) = 0, andBi =hc RkFo =atR2q=T - T¥To - T¥The common definition of Bessel’s functions applies here.For the similar situation involving a solid sphere, the following holds:¥q=2sin(l n R) - l n R cos(l n R)å l R - sin(l R) cos(l R) exp(-l R Fo)n =1nn2nn2sin(l n r )l nr(4.1.30)and the ln are found as the roots of lnR cos lnR = (1 – Bi) sin lnR. Otherwise, the same definitions aswere given for the cylinder hold.Solids that can be envisioned as the geometric intersection of the simple shapes described above canbe analyzed with a simple product of the individual-shape solutions.
For these cases, the solution isfound as the product of the dimensionless temperature functions for each of the simple shapes withappropriate distance variables taken in each solution. This is illustrated as the right-hand diagram inFigure 4.1.4. For example, a very long rod of rectangular cross section can be seen as the intersectionof two large plates. A short cylinder represents the intersection of an infinitely long cylinder and a plate.The temperature at any location within the short cylinder isq 2 R,2 L Rod = q Infinite 2 R Rod q 2 L Plate(4.1.31)Details of the formulation and solution of the partial differential equations in heat conduction arefound in the text by Arpaci (1966).Finite-Difference Analysis of ConductionToday, numerical solution of conduction problems is the most-used analysis approach.
Two generaltechniques are applied for this: those based upon finite-difference ideas and those based upon finiteelement concepts. General numerical formulations are introduced in other sections of this book. In thissection, a special, physical formulation of the finite-difference equations to conduction phenomena isbriefly outlined.Attention is drawn to a one-dimensional slab (very large in two directions compared with thethickness). The slab is divided across the thickness into smaller subslabs, and this is shown in Figure4.1.5. All subslabs are thickness Dx except for the two boundaries where the thickness is Dx/2.
Acharacteristic temperature for each slab is assumed to be represented by the temperature at the slabcenter. Of course, this assumption becomes more accurate as the size of the slab becomes smaller. With© 1999 by CRC Press LLC4-11Heat and Mass TransferFIGURE 4.1.4. Three types of bodies that can be analyzed with results given in this section. (a) Large plane wallof 2L thickness; (b) long cylinder with 2R diameter; (c) composite intersection.FIGURE 4.1.5.
A one-dimensional finite differencing of a slab with a general interior node and one surface nodedetailed.the two boundary-node centers located exactly on the boundary, a total of n nodes are used (n – 2 fullnodes and one half node on each of the two boundaries).In the analysis, a general interior node i (this applies to all nodes 2 through n – 1) is considered foran overall energy balance. Conduction from node i – 1 and from node i + 1 as well as any heat generationpresent is assumed to be energy per unit time flowing into the node. This is then equated to the timerate of change of energy within the node. A backward difference on the time derivative is applied here,and the notation Ti ¢ ; Ti (t + Dt) is used. The balance gives the following on a per-unit-area basis:Ti -¢ 1 - Ti ¢ Ti +¢ 1 - Ti ¢T ¢- Ti++ q˙ G,i Dx = rDxc p iDx kDx k +Dt(4.1.32)In this equation different thermal conductivities have been used to allow for possible variations inproperties throughout the solid.The analysis of the boundary nodes will depend upon the nature of the conditions there.
For thepurposes of illustration, convection will be assumed to be occurring off of the boundary at node 1. Abalance similar to Equation (4.1.32) but now for node 1 gives the following:© 1999 by CRC Press LLC4-12Section 4T¥¢ - T1¢ T2¢ - T1¢DxDx T ¢ - T++ q˙ G,1= r cp 1 11 hcDx k+22Dt(4.1.33)After all n equations are written, it can be seen that there are n unknowns represented in theseequations: the temperature at all nodes. If one or both of the boundary conditions are in terms of aspecified temperature, this will decrease the number of equations and unknowns by one or two, respectively.
To determine the temperature as a function of time, the time step is arbitrarily set, and all thetemperatures are found by simultaneous solution at t = 0 + Dt. The time is then advanced by Dt and thetemperatures are then found again by simultaneous solution.The finite difference approach just outlined using the backward difference for the time derivative istermed the implicit technique, and it results in an n ´ n system of linear simultaneous equations. If theforward difference is used for the time derivative, then only one unknown will exist in each equation.This gives rise to what is called an explicit or “marching” solution. While this type of system is morestraightforward to solve because it deals with only one equation at a time with one unknown, a stabilitycriterion must be considered which limits the time step relative to the distance step.Two- and three-dimensional problems are handled in conceptually the same manner.
One-dimensionalheat fluxes between adjoining nodes are again considered. Now there are contributions from each of thedimensions represented. Details are outlined in the book by Jaluria and Torrance (1986).Defining TermsBiot number: Ratio of the internal (conductive) resistance to the external (convective) resistance froma solid exchanging heat with a fluid.Fin: Additions of material to a surface to increase area and thus decrease the external thermal resistancefrom convecting and/or radiating solids.Fin effectiveness: Ratio of the actual heat transfer from a fin to the heat transfer from the same crosssectional area of the wall without the fin.Fin efficiency: Ratio of the actual heat transfer from a fin to the heat transfer from a fin with the samegeometry but completely at the base temperature.Fourier’s law: The fundamental law of heat conduction.
Relates the local temperature gradient to thelocal heat flux, both in the same direction.Heat conduction equation: A partial differential equation in temperature, spatial variables, time, andproperties that, when solved with appropriate boundary and initial conditions, describes thevariation of temperature in a conducting medium.Overall heat transfer coefficient: The analogous quantity to the heat transfer coefficient found inconvection (Newton’s law of cooling) that represents the overall combination of several thermalresistances, both conductive and convective.Thermal conductivity: The property of a material that relates a temperature gradient to a heat flux.Dependent upon temperature.ReferencesAbramowitz, M.
and Stegun, I. 1964. Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55.Arpaci, V. 1966. Conduction Heat Transfer, Addison-Wesley, Reading, MA.Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, 2nd ed., Oxford University Press,London.Guyer, E., Ed.
1989. Thermal insulations, in Handbook of Applied Thermal Design, McGraw-Hill, NewYork, Part 3.Jaluria, Y. and Torrance, K. 1986. Computational Heat Transfer, Hemisphere Publishing, New York.Sparrow, E. 1970. Reexamination and correction of the critical radius for radial heat conduction, AIChEJ. 16, 1, 149.© 1999 by CRC Press LLCHeat and Mass Transfer4-13Further InformationThe references listed above will give the reader an excellent introduction to analytical formulation andsolution (Arpaci), material properties (Guyer), and numerical formulation and solution (Jaluria andTorrance). Current developments in conduction heat transfer appear in several publications, includingThe Journal of Heat Transfer, The International Journal of Heat and Mass Transfer, and NumericalHeat Transfer.© 1999 by CRC Press LLC4-14Section 44.2 Convection Heat TransferNatural ConvectionGeorge D.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
