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Essentials of Chemical Reaction Engineering, 2nd Edition by H. Scott Fogler

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12.7.3 Energy Balance

Using the energy flux, e, to carry out an energy balance on our annulus (Figure 12-15) with system volume 2πrΔrΔz, we have

(Energy flow in at r) = er Acr = er · 2πrΔz

(Energy flow in at z) = ez Acz = ez · 2πrΔr

(Energy0flowin0at0r)-(Energy0flowout0at0r+Δr)+(Energy0flowin0at0z)-(Energy0flowout0at0z+Δz)=(Accumu1ationof0energy0invo1ume0(2πrΔrΔz))

(er2πrΔz)|r-(er2πrΔz)|r+Δr+eZ2πrΔr|Z-eZ2πrΔr|z+Δz0=00

Dividing by 2πrΔrΔz and taking the limit as Δr and Δz → 0

1r(rer)reZz=0(12-49)

The radial and axial energy fluxes are

er0=0qr+ΣWirHiez0=0qz+ΣWizHi

Substituting for the energy fluxes, er and ez

-1r[r[qr0+0ΣWir0Hi]]r-[qz+ΣWiz0Hi]

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