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Derive a relation of heat transfer through fin with heat loosing tip.
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Consider a volume element of a fin at location x having a length dx, C/S area Ac and a perimeter P as shown in fig under steady state condition,

Energy equation for above fig can be written as

(rateofheatconductionintotheelementatx)=(rateofheatconductionfromtheelementatx+dx)+(rateofheatconvectionfromtheelement)

enter image description here

Fig 2.1: Volume element of a fin at location x having a length of dx.

Qx=Q(x+dx)+Qconv

Q(x+dx)Qxdx+hp(tta)=0………(i) as Qconv=h(pdx)(tta)

dQxdx+hP(tta)=0

Qcond=kAc dtdx………(ii)

Substituting the value of Qcond from the above equation in eq (i), we get the differential equation governing heat transfer in fins:

ddx (kAc dtdx)-hP(tta)=0………(iii)

d2tdx2-hPkAc(tta)=0

d2tdx2-m2θ=0………(iv)

Where m=hPAck……….(v)

θ=tta

Solving equation (iv) we will get the solution as follows

θ=C1emx+C2e(mx)………(vi)

To solve above equation we will apply boundary condition. boundary conditions are heat dissipation from a fin losing heat at the tip.

enter image description here

Fig2.1: fin of finite length losing heat at the tip

Please note TL=t, T=ta

Boundary condition can be obtained by energy balance at fin tip.

kAcdtdx(x=L)=hAs(tta)

kdθdx(x=L)=hθ(x=L) ………(vii)

Substituting above equation in (vi) we get,

-km[C1e(mL)+C2emL]=h[C1e(mL)+C2emL]

Rearranging as,

C1e(mL)C2emL=hmk[C1e(mL)+C2emL]

Solving above

θ0e(mL)C2e(mL)C2emL=h/mk[θ0e(mL)C2e(mL)+C2emL] asθ0=C1+C2 from eq (vi)

Solving above we will get,

C2=(θ0e(mL)hmke(mL)emL+e(mL)+hmk[emLe(mL)

Similarly, we get

C1=θ0emL+hmkemLemL+e(mL)+hmk[emLe(mL)]

Substituting C1 and C2 in equation (vi) we get,

θ(x)θ0 =(em(Lx)+hmkem(Lx)+e(m(Lx))hmkem(Lx))(emL+e(mL)+hmk[emLe(mL)])

It may be rearranged and expressed in terms of hyperbolic functions as

θ(x)θ0 =t(x)ta)(tta)=(coshm(Lx)+(hmk)sinhm(Lx))(cosh(mL)+(hmk)sinh(mL))

The total heat transfer from fin

Q0=Qfin=kAc(dθdx)(x=0)=

(hPkAc)(tta)(sinh(mL)+hmkcosh(mL))(cos(mL)+hmksinh(mL))

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