`I. Analytical Solutions in Conduction Heat TransferC. Lumped formulation of first law of thermodynamics&quot;A general law is said to be lumped if its terms are independent of space and to be distributed if its terms depend on space.&quot; Arpaci (1966)    dEcv V2 V2 = Qcv + Wnonflow +  mi  h + + gz  -  me  h + + gz      dt 2 2 in  i out  e(I.C.1)1 where E = U + mV 2 + mgz 2dmcv =  mi -  me (kg / s) dt in outlesson 4m = VA =VA (kg / s) v(I.C.2)I. Analytical Solutions in Conduction Heat TransferC. Lumped formulation of first law of thermodynamics (cont.)If we consider an incompressible substance at constant pressure and neglect viscous forces, shaft work, KE and PE, (I.C.1) becomesc dT =  mi c i (Ti - Tref ) -  me ce (Te - Tref ) +  q j A j + u' ' ' dt i e j(I.C.3)where · qj is heat flux normal to area Aj at the control volume boundary · TR is a reference temperaturelesson 41I. Analytical Solutions in Conduction Heat TransferD. Inductive versus deductive approach1. Analysis of triangular straight fin. We want to find T(x) and heat loss from fin of width w. The drawing is not to scale. It is important to realize that L &gt;&gt; b.A' = bx LFluid temperature = T  hqx = -kdT dxbqC = hC (T - T )h LxBase temperature = Tblesson 4I. Analytical Solutions in Conduction Heat Transfer1. Analysis of triangular straight fin (cont.) Assume no T gradients in y-direction. Assume perimeter of fin is P = 2 dx (fin approximation). Assume k = constant. Energy balance per unit width of fin giveskb d  dT  x  - 2h(T - T ) = 0 L dx  dx Boundary conditions: at x = 0, T(0) = finite at x = L, T(L) = Tblesson 4(I.D.1)(I.D.2) (I.D.3)2I. Analytical Solutions in Conduction Heat Transfer2. Solution The solution is a modified Bessel function of the first kind of order zero.I 2c x T - T 2hL where c 2 = = 0 Tb - T I0 2c L kbThe heat rate (W) is( () )(I.D.4)Q c L I1 2c L = where A = bw kA (Tb - T ) I0 2c L L(())(I.D.5)lesson 4I. Analytical Solutions in Conduction Heat Transfer3. Deductive approach with (I.B.18) 0 T ~  c  + V  T  =   (kT ) + u' ' '  t I.B.18 does not fit our geometry nor does it accommodate a mixed formulation.lesson 43`

3 pages

#### Report File (DMCA)

Our content is added by our users. We aim to remove reported files within 1 working day. Please use this link to notify us:

Report this file as copyright or inappropriate

1225011

### You might also be interested in

BETA 