2 Nature and Rationale of Extended Surfaces An extended surface (also know as a combined conduction-convection systemor a fin) is a solid within which heat transfer by conduction is assumed to beone dimensional, while heat is also transferred by convection (and/orradiation) from the surface in a direction transverse to that of conduction.If heat is transferred from the surface to the fluid by convection, whatsurface condition is dictated by the conservation of energy requirement?Why is heat transfer by conduction in the x-direction not, in fact, one-dimensional?
3 Nature and Rationale (cont.) What is the actual functional dependence of the temperature distribution inthe solid?If the temperature distribution is assumed to be one-dimensional, that is,T=T(x) , how should the value of T be interpreted for any x location?How does vary with x ?When may the assumption of one-dimensional conduction be viewed as anexcellent approximation?The thin-fin approximation.Extended surfaces may exist in many situations but are commonly used asfins to enhance heat transfer by increasing the surface area available forconvection (and/or radiation).They are particularly beneficial when is small,as for a gas and natural convection.Some typical fin configurations:Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annularfin, and (d) pin fin of non-uniform cross section.
4 Heat Transfer from Extended Surfaces Extended Surface to Increase Heat Transfer RateHeat Transfer from Extended Surfaces
5 Heat Transfer from Extended Surfaces Extended Surface to Increase Heat Transfer RateThree ways to increase qIncrease hReduceIncrease A
6 Heat Transfer from Extended Surfaces Applications of FinsHeat Transfer from Extended Surfaces
7 Heat Transfer from Extended Surfaces A General Conduction AnalysisHeat Transfer from Extended Surfaces
8 Heat Transfer from Extended Surfaces Applying Conservation of energy to the differential element:From Fourier’s Law
9 Heat Transfer from Extended Surfaces From Taylor Series Expansion:Heat of Convection
10 Heat Transfer from Extended Surfaces Heat Transfer Eqn:Or
12 Fins of Uniform Section Area Heat Transfer Eqn:Where dAs/dx = P (fin perimeter), dAC/dx = 0Define Excess Temperature
13 Fins of Uniform Section Area Define:Heat Transfer Eqn.:General Solution
14 Fins of Uniform Section Area Need Two b.c.’s to evaluate C1 and C2.Fins of Uniform Section Area
15 Fins of Uniform Section Area b.c.-1:b.c.-2:Or b.c.-2From b.c.-1:From b.c.-2:
16 Fins of Uniform Section Area Solving for C1 and C2:The hyperbolic function definitions (Page 1014):
17 Fins of Uniform Section Area Overall Heat Transfer Rate:Above equation can be obtained from energy conservation of the fin.
18 Fins of Uniform Section Area If the fin tip is adiabatic (second tip b.c.)Fins of Uniform Section Area
19 Fins of Uniform Section Area If the fin tip is at constant T (Third tip b.c.)Fins of Uniform Section Area
20 Fins of Uniform Section Area If the fin is very long (Fourth tip b.c.)Fins of Uniform Section Area
21 Fins of Uniform Section Area Tip Condition(x=L)T distributionFin Heat TransferABCD
22 Example 3.9 (pages )A very long rod 5 mm in diameter has one end maintained at 100 ºC. The surface of the rod exposed to ambient air at 25 ºC with a convection heat transfer coefficient of 100 W/m2 K.1. Determine the temperature distributions along rods constructed from pure copper, 2024 aluminum alloy, and type AISI 316 stainless steel. What are the corresponding heat losses from the rods?2. Estimate how long the rods must be for the assumption of infinite length to yield accurate estimate of the heat loss.
23 Example 3.9 Known: A long circular rod exposed to ambient air Find: 1. T distribution and heat loss when rod is fabricated from copper an aluminm alloy, or stainless steel2. How long rods must be to assume infinite lengthSchematic:
24 Example 3.9 Assumptions: Properties: Steady-state; 1-D conduction in x direction;Constant properties.Negligible radiation;Uniform heat transfer coefficient;Infinite long rodProperties:
29 Example 3.9Comments:The mL 2.65 was based on heat loss. If the requirement is to predict the T distribution accurately, a larger mL value (4.6) is needed. It was based on exp(-mL)<0.01.
30 Problem 3.126Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at T=1200 ºC and maintains a convection coefficient of h=250 W/m2 K over the blade. The blades, which are fabricated from Inconel, k 20 W/m K, have a length of L=50 mm. The blade profile has a uniform cross-sectional area of A=6xl0-4m2 and a perimeter of P =110 mm. A proposed blade-cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of T = 300 ºC.
31 Problem 3.126(a). If the maximum allowable blade temperature is 1050 ºC and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory?(b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant?
32 Problem: Turbine Blade Cooling Problem 3.126: Assessment of cooling scheme for gas turbine blade.Determination of whether blade temperatures are lessthan the maximum allowable value (1050°C) forprescribed operating conditions and evaluation of bladecooling rate.Schematic:Assumptions: (1) One-dimensional, steady-state conduction in blade, (2) Constant k, (3)Adiabatic blade tip, (4) Negligible radiation.Analysis: Conditions in the blade are determined by Case B of Table 3.4.(a) With the maximum temperature existing at x = L, Eq yields
33 Problem: Turbine Blade Cooling (cont.) From Table B.1 (or by calculation), Hence,and, subject to the assumption of an adiabatic tip, the operating conditions are acceptable.Eq and Table B.1 yieldHence,<Comments: Radiation losses from the blade surface contribute to reducing the bladetemperatures, but what is the effect of assuming an adiabatic tip condition? Calculatethe tip temperature allowing for convection from the gas.