1 Chapter 2: Steady-State One-Dimensional Heat Conduction 2.1 …………. Conduction through a plane wall & composite wall2.2 …………. Conduction from fluid to fluid through a composite wall2.3 …………. Overall Heat Transfer Coefficient2.4 …………. Conduction through a hollow cylinder & composite cylinder2.5 …………. Conduction through a sphere2.6 …………. Boundary and Initial Condition2.7 …………. Critical thickness of Insulation
2 2.1 … Steady State Conduction through a plane wall T2 T1xQ.kLThermal resistance (in k/W)(which opposing heat flow)1/22/05ME 259
3 Thermal Analogy to Ohm’s Law Electrical AnalogyThermal Analogy to Ohm’s LawOHM’s LAW :Flow of ElectricityV=IR electTemp Drop=Heat Flow × ResistanceVoltage Drop = Current flow ×Resistance
4 Steady State Conduction through a composite wall Q.ΔxAABCΔxBΔxCT1Q.T2T3kAkBT4kAkBkCkCx
5 Steady State Conduction through a composite wall
6 1 D Heat flow from fluid to fluid through plain wall Ts,1Ts,2T∞,2……kx=0x=LHot fluidCold fluidqxAh21AkLAh1(Thermal Resistance )
7 2.2…1-D Heat flow from fluid to fluid through composite wall BCT∞,1T∞,2h1h2K AK BK CL AL BL Cq xT∞,1T∞,2Ah12
8 Parallel composite The heat transfer rate in the network is parallel Alternatively, the heat transfer rate can be calculated as the sum of heat transfer rates in the individual materials, i.e.
9 Series composite Contact Resistance In composite systems, the interface between two layers is usually not perfect. This is due to surface roughness effect.Contact spots between the two layers are interspersed with gaps that are, in most instances, air filled.The additional resistance between the two layers, called thermal contact resistance Rt, results in temperature drop across the interface.Thermal contact resistance is dependent upon the solid materials, surface roughness, contact pressure, temperature, and interfacial fluid.
10 2.3 … Overall Heat Transfer Coefficient A modified form of Newton’s Law of Cooling to encompass multiple resistancesto heat transfer.In above equation,Overall heat transfer coefficient
11 2.4 … Conduction through hollow cylinder Consider a hollow cylinder of length L (shown below), whose inner and outer surfaces are exposed to fluids at different temperatures. The system is analyzed by the standard method as follows:
12 Conduction through hollow cylinder (continue….) For steady-state conditions with no heat generation, the heat equation for the system isThe boundary conditions areIntegrating the heat equation (assuming constant k) and using the boundary conditions yieldTherefore, the temperature distribution associated with radial conduction through a cylindrical wall is logarithmic.The heat transfer rate is obtained by using the temperature distribution with Fourier’s law:where,
13 Conduction through hollow cylinder (continue….) For pure conduction, we considered resistances in series in which our primaryinterest was in the temperatures at the inner and outer walls (not the interior walls).For convection, a similar principle arises.Consider a pipe filled with hot fluid at temperature T1. Define intermediatetemperatures as follows:T1T2InsulationAir T5Hot liquidT3PipeT4
14 Conduction through Composite Cylinder Consider a composite cylindrical wall of length L shown below.
15 Conduction through Composite Cylinder (continue) Assumption: Neglecting interfacial contact resistancesThe heat transfer rate may be expressed asAbove equation may be expressed in terms of an overall heat transfer coefficient asIf U is defined in terms of the inside area A1, above equations may be equated to yieldNote: Similar equations could be written for U2, U3, etc.
16 Conduction through Composite Cylinder (continue) For pure conduction, we considered resistances in series in which our primaryinterest was in the temperatures at the inner and outer walls (not the interior walls).For convection, a similar principle arises.Consider a pipe filled with hot fluid at temperature T1. Define intermediatetemperatures as follows:T1T2InsulationAir T5Hot liquidT3PipeT4
17 The heat transfer rate for each "step": q1->2 = conduction or convection?q2->3 = conduction or convection?q3->4 = conduction or convection?q4->5 = conduction or convection?ConvectionConductionConductionConvection
21 With U known, it is a simple matter to calculate the overall heat transfer rate given the total temperature difference.Note that we have used the symbol Uo because the overall heattransfer coefficient was defined with respect to the outside area(Ao) of the pipe. This is the most common practice.We could equally have defined 1/Ui or UiAi. In this case, wewould use Ai as the basis for calculations. In either case, thevalues of U would be slightly different, but UA and hence q arethe same.
22 Equation for 1/UiRule of thumb: If the ratio of Do/Di is less than 1.5, then the arithmetic average ofDo and Di is roughly equal to the log mean average (good for ANY log mean average).Limiting Resistance:A very useful concept in heat transfer is that of limiting resistance.What is limiting resistance?What would the limiting resistance be for a hot liquid flowing inside an uninsulatedpipe?How would you increase the heat transfer rate?
23 2.5…Conduction through Sphere Heat EquationTemperature Distribution for Constant k:
24 Heat flux, Heat Rate and Thermal Resistance: Composite Shell:
25 2.6… Boundary and Initial Conditions Heat equation is a differential equation:Second order in spatial coordinates: Need 2 boundary conditionsFirst order in time: Need 1 initial conditionBoundary ConditionsB.C. of first kind (Dirichlet condition):xT(x,t)Tsx=0Constant Surface TemperatureAt x=0, T(x,t)=T(0,t)=Ts
26 Boundary and Initial Conditions (Continue…) B.C. of second kind (Neumann condition): Constant heat flux at the surfaceFinite heat fluxxT(x,t)qx”= qs”Adiabatic surfaceqx”=0xT(x,t)
27 Boundary and Initial Conditions (continue…) 3) B.C. of third kind: When convective heat transfer occurs at the surfaceT(x,t)T(0,t)x= q”conductionq”convection
28 2.7…Critical Radius of Insulation We know that by adding more insulation to a wall always decreases heat transfer.This is expected, since the heat transfer area A is constant, and adding insulation will always increase the thermal resistance of the wall without affecting the convection resistance.However, adding insulation to a cylindrical piece or a spherical shell, is a different matter.The additional insulation increases the conduction resistance of the insulation layer but it also decreases the convection resistance of the surface because of the increase in the outer surface area for convection.Therefore, the heat transfer from the pipe may increase or decrease, depending on which effect dominates.A critical radius (rcr) exists for radial systems, where:adding insulation up to this radius will increase heat transferadding insulation beyond this radius will decrease heat transferFor cylindrical systems, rcr = kins/hFor spherical systems, rcr = 2kins/h
29 Critical Radius of Insulation (continue….) Consider a cylindrical pipe, where, r1 -- outer radius T1 -- constant outer surface temperature k -- thermal conductivity of the insulation r2 -- outer radius - temperature of surrounding medium h - convection heat transfer coefficient Insulated Cylindrical Pipe
30 Critical Radius of Insulation (continue….) The rate of heat transfer from the insulated pipe to the surrounding air can be expressed as The variation of heat transfer rate with the outer radius of insulation r2 is plotted in Figure The value of r2 at which heat transfer rate reaches maximum is determined from the requirement that (zero slope). Performing the differentiation and solving for r2 gives us the critical radius of insulation for a cylindrical body to be NOTE: The rate of heat transfer from the cylinder increases with the addition of insulation for r2< rcr, reaches a maximum when r2= rcr, and starts to decrease for r2> rcr. Thus, insulating the pipe may actually increase the rate of heat transfer from the pipe instead of decreasing it when r2< rcr .
31 Critical Radius of Insulation (continue….) good for steam pipes etc.good for electrical cablesR c r=k/hr0R t o tVariation Of Heat Transfer Rate With Radius
32 Temperature Distribution One-Dimensional Steady State Solutions to the Heat Equation With No GenerationPlane WallCylindrical WallSpherical WallHeat EquationTemperature DistributionHeat Flux (q” )Heat Rate (q)Thermal Resistance(Rt, cond)