Fins equation & lumped heat capacity system

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    29-Nov-2014
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Two basic topics of heat transfer have been covered up by me based on the famous books of :- 1) John H. Lienhard (Professor Emeritus, University of Houston) 2) J.P. Holman (Professor, Southern Methodist University) 3) Prabal Talukdar (Associate Professor, IIT, India)

Transcript of Fins equation & lumped heat capacity system

  • 1. SMYTH_WORKS FINS INS EQUATION, UNSTEADY C UMPED HEAT CAPACITY SYSTEM LUMPED Das Buch ist schlecht und CONDITION & YSTEM die Lehrer ist wirklich Streber 9/1/2014 Streber, auch!
  • 2. 2 General Equations for a One-Dimensional Fin: We take a consideration that, this is a steady-state heat transfer process. Heat flows through an elemental cross-section. Where, x = Length of the cross-section AS = Surface area AC = Cross-sectional area h = Heat transfer coefficient Tf = Temperature of the fluid. Convection occurs at the surface and hereby, writing down the heat-balance equation in words: Heat Flow (into element) = Heat flow (out of element) + Heat transfer (into surroundings) Or, QX = QX+X + h. AS .( T-Tf ) --------------------------------------------------------------(i) From Fouriers Law: QX = -kAC --------------------------------------------------------------------------------------------- (ii) From Taylors Series, using equation (ii) we get: QX+X = QX + Smyth_Works (-kAC ) x ------------------------------------------------------------------ (iii) So, combining equation (i) & (iii), this becomes: (kAC ) x - h. AS .( T-Tf ) = 0 -----------------------------------------------------------(iv) The left-sided term is identical to the result for a plane wall. The difference here is that the area is not constant with x.
  • 3. 3 So, using the product-rule to multiply out the left-sided term, gives us: kAC . + k. . - . .(T-Tf ) = 0 + Smyth_Works . . -
  • 4. . . .(T-Tf ) = 0 ---------------------------------------------(v) From figure, AS = P.x [here, x = length of the whole fin; P = perimeter of the fin] = P -----------------------------------------------------------------------------------------------------(vi) Putting the value of equation (vi) into (v), we get: + . . -
  • 5. . . .(T-Tf ) = 0 -
  • 6. . . .(T-Tf ) = 0 -------[Let, the fins has uniform cross-section; so, 0]------(vii) - m2 = 0 ; Let, m=
  • 7. . . and, =( T-Tf ). It is called the general equation for one-dimensional fins. Solution of - m2 = 0 : General solution of the above equation is: = C1e mx + C2e mx --------------------- [C1 & C2 = Constants; depend on the boundary condition] O = (C1+C2) -------------------------------[But, C2 = 0 and thus C1 = O ] = O .e mx = e mx ------------------------------------------------------------------------------------------------(viii) = e mx Therefore, = = e mx ; it is the solution for one-boundary condition.
  • 8. 4 Smyth_Works Solutions for above equation in a Tabular form Criteria Boundary Condition Solution Heat Transfer Case 001 (i) Fin is very long (x= ) (ii) Tend of fin=Tfluid (surrounding fluid) at, x= 0 = 0 at, x= = 0 = = e mx q = PA . Case 002 (i) Fin is of finite length (ii) Loses heat by convection from its end. N/A (Holman-p43) Case 003 Tend of fin = insulated at, x= 0 = 0 at, x= L = 0 = !"# [%(')] !"#(%') q = PA ..tanh (mL) Fin efficiency = *+,-./ 01., 2.345121 61./ 01., 2.345121 (01., 789+8 7:-/ ;1 ,2.345121) = f If the entire area is at base temperature, then, f = * .?.,.38 (@A)
  • 9. 5 Unsteady State Condition : When, a solid body suddenly subjected to a change in environment, sometimes elapse belong an equilibrium temperature. Condition will prevail; this is called transient problem. Mathematically, D = E DE Smyth_Works = F G K . L [MN I JI O ]PQ .sinIG R' where, n = 1, 3, 5, up to (2n+1)th term. Equation for Lumped Heat Capacity System (LHCS) : Lumped heat capacity system assumes that, resistance of heat conduction is so small compared to the resistance of heat convection. Mathematically, Rcond. 0 then, T = T and, T - T = T0 - T Then, integrating equation (v) we get: ln ( ZZT ZZT ) = - #[. ]^_ ( ZZT ZZT ) = e ab.c def --------------------------------------------------------------- (vi) Introducing new dimensionless temperature, = ( ZZT ZZT ) And time constant, = ]^_
  • 12. Rewritten equation (vi) will be like this, = Lg h .
  • 13. 7 Applicability of LHCS : Biot number, Bi = Smyth_Works (O i) (O j) = klmnopqrlm sturuqvmpt klmwtpqrlm sturuqvmpt =
  • 14. (x y) [Bi< 0.1] where, z = Characteristic length = L. Transient Heat Flow in a Semi-Infinite Solid : Let us consider, there is a semi-infinite solid shown in figure beside, maintained at some initial temperature = Ti , suddenly lowered surface temperature = T0 . So, the differential equation for temperature distribution T(x,) is: {Z { = P.{Z {| ----------------------------------------------- (i) The boundary conditions are: T(x,0) = Ti T(0,) = T0 [for > 0] Then the solution of equation (i) will be: (,|) Z D = erf RPQ Where, the Gauss error function is defined as: RPQ = R G /RPQ L erf .d Here, = Dummy variable (i.e. = x, y, z, etc)
  • 15. 8 Smyth_Works ________________