Question

The heat that is conducted through a body must frequently be removed by other heat transfer processes. For example, the heat generated in an electronic device must be dissipated to the surroundings through convection by means of fins. Consider the one-dimensional aluminum fin (thickness t = 3.0 mm, width Z = 20 cm, length L) shown in Figure 1, that is exposed to a surrounding fluid at a temperature T. The conductivity of the aluminum fin (k) and coefficient of heat convection of surrounding fluid (h) are given in Figure 1. The defining equation for the conduction and convection heat-transfer coefficients are recalled as qcond. = -k A dT / dx and qconv. = h A’ (T − T) where the area in these equations (A and A’) are the surface area for conduction and convection, respectively.

a)

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

v

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

v

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

1. h = 10 W/m2 · ◦C

   The temperature of the base of the fin is T₀. Make an energy balance on the element of the fin of thickness dx as shown in the figure 1 and develop a linear ODE as the mathematical model for this heat transfer problem.

Answer 1

Convehon to aat Too& hmi Base wall at To restoy &he in NeL tohne A- Prefile rea = (2Kt) L = Leagth of fin =th'c kness caf h objechive1 Toget Kempenatune Diyhibution un'thin the fin ie T-fln) TO gut Hest trarfen Rate though entire fine #AMume Stale Heat tranypn cordi hon e Tf(he) diffenendtl elament of he fin of Jonyth d at Conrden a rem the rost ofe fin a oltance of uhene the the fin is T 7 heat conolsctd into the ebnont -kA dT het da Heat conaluckd out of he elament td (9)d

neat Cenvectcol from the Saface of elament to flud h Pdr (T-T) Ennyy balance e wheare pa permeter n=9Arda t Convecked + h Pdn (T-To) hPoa (T-Tao) -kA dT he (T-To) 0 kA e fln) d'Tdo put het T-Tao Ph m = do 1KA 11 Thes i the Standandl for mut 2ndorden dittrenh'al eg of dirett ghey So uhene the sal com Cre +C2e Ph kA whene m- coral'hon s beundary One t enmon / T To d00, (To-Too) at neo i.e at the root of fin

en three difdennt cae Secend beundary Condlhon daperal of fin () Case I Fin is Infinitely Jong Than the temp at the tip of the fin will be esentilly that the ambient fin, 1.e at nE 0 ) T=Too , than -0 Then Sol of temp Distnibuhon fr infintelz lory fin Cate is -mn T-Too Ph Too Nole Trate thengh enhe fh For any fin Care haat trarten nle be equae to heat Cenoluekd into dhe fin atih Le Hest coroluckol i'nho he fn at 1 rest --kA dn latheo (T.-Too 9frPh KA

Fin is finite langtn cad-2 but its tip is Insulateal . Heat Conelueteol 1nto the hp of fhso cLT dn heL dh/hel Then T-To Coshm (L-n) To-Tao Cosh m2 9 fin ph KA CTo- To) dem h m L but tip ionot Inulateel Ccrc2-Fine o inte in than the Jal fr Temp Dishnbuton nn the 1 . fin T-To (esh m Coshmle To-Too Whene Le Corected length of fn 9fin=PAKA (To-T)tenhm (Lt) for Pacteng tan fin (Lt Pin fin