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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)
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(t−ta)=0………(i) as Qconv=h(pdx)(t−ta)
dQxdx+hP(t−ta)=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(t−ta)=0………(iii)
d2tdx2-hPkAc(t−ta)=0
d2tdx2-m2θ=0………(iv)
Where m=√hPAck……….(v)
θ=t−ta
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.
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(t−ta)
−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[emL−e(−mL)
Similarly, we get
C1=θ0emL+hmkemLemL+e(−mL)+hmk[emL−e(−mL)]
Substituting C1 and C2 in equation (vi) we get,
θ(x)θ0 =(em(L−x)+hmkem(L−x)+e(−m(L−x))−hmke−m(L−x))(emL+e(−mL)+hmk[emL−e(−mL)])
It may be rearranged and expressed in terms of hyperbolic functions as
θ(x)θ0 =t(x)−ta)(t−ta)=(coshm(L−x)+(hmk)sinhm(L−x))(cosh(mL)+(hmk)sinh(mL))
The total heat transfer from fin
Q0=Qfin=−kAc(dθdx)(x=0)=
√(hPkAc)(t−ta)(sinh(mL)+hmkcosh(mL))(cos(mL)+hmksinh(mL))