HEAT TRANSFER: CONDUCTION AND CONVECTION

HEAT TRANSFER: CONDUCTION AND CONVECTION

  Dr. B. Jayaraman _{FDP on CFD} Lecture 5: 7 ^{th June 2016,}

  SoME, SASTRA University, Thanjavur OUTLINE

• Introduction
• Conduction • Convection

   Forced Convection  Free Convection

• Other Models • Conclusion

  MODES OF HEAT TRANSFER

  THERMAL CHARACTERISTICS Temperature Distributions Amount of Heat Lost or gained

  Thermal Analysis Thermal Gradients Thermal Fluxes Thermal Analysis

• + Thermal Stresses Stress Analysis

PHENOMENA (MODELS)

• Conduction • Convection

   Forced Convection  Natural convection  Boiling (Multiphase)

• Radiation • Species diffusion and Combustion • Conjugate heat transfer
• Periodic heat transfer

BOUNDARY CONDITIONS

• Heat Flux • Temperature • Convection • Radiation • Mixed – Combination of Convection and

  Radiation boundary conditions

• Via System Coupling

  PROPERTIES _•

Fluid properties such as heat capacity, conductivity and viscosity

can be defined as:

   Constant  Temperature-dependent  Composition-dependent  Computed by kinetic theory  Computed by user-defined functions

• Density can be treated as

   Constant (with optional Boussinesq modeling)  Temperature-dependent  Computed as ideal gas law  Composition-dependent  User defined functions

  OUTLINE

• Introduction
• Conduction
• Convection

   Forced Convection  Free Convection

• Other Models • Conclusion

  CONDUCTION _{conducted Generated}

• _{+ Energy Heat + = conducted Energy Internal change of} Rate of into CV within CV _{out of CV Energy in CV}

  2 ^{2 2} c

    ρ

  T T T q T

   

_p  ₂     _{2 2} 

   x y z k k t

  

   

 

2-DIM. PROBLEM

  

Physical domain of Slab with square cross section

^  T T ^  T T

    T

  T ^  T T

  2L

^

^

^

^{   W 3 m q 2-dim, Steady state with heat generation 2 2 2 2}   

       

      ^T _k ^q _{y x} ^T 

  2-DIM,STEADY STATE _{T  T x y}  _{Geometrically and thermally symmetric θ}  ; _{q L L L 2} ^{X  ; Y } ₂  _{k 2} T  T _

      θ θ

  L

  1 ₂    ₂    X  Y _{Non-dim Boundary Conditions}  

  W _{q  }   ₃ T  T _d  T _m ^ _{At X = 0} θ _

    _x  _dX  _{At X = 1 } θ _{d T At Y = 0 dY} θ _

   L _y  _{At Y = 1}  

  θ

Computational Domain of Slab with square cross section

  2-DIM,UNSTEADY STATE SQUARE PLATE _{T  T x y t i} α  ; ^{X  ; Y  } _{Infinitely long plate, initially at T Then suddenly maintained at T o i} θ _{T  T L L L o i} τ _{2 2 2}

  2 ²     

  θ θ θ      _{T T T 2}   ₂

    α _{2 2}

     

   X  Y  τ

   x  y  t  

    _{Non-dim Boundary Conditions T T}  _{o At  τ= 0} θ

  L  _{1  T T T  o For At X = 1 τ> 0 d} θ _{ x  At X = 0 d dX} θ _{  T  L At Y = 0 dY} θ _{ } y ^{At Y = 1} ₁

   θ OUTLINE

• Introduction • Conduction
• Convection

   Forced Convection  Free Convection

• Other Models • Conclusion

  CONVECTION

• Diffusion Transport + Advection Transport

  driver

• Advecting variable(velocity) 

  passenger

• Advected variable(temperature)

  

• Transports Momentum and Energy

FORCED CONVECTION

  Diffusion Transport

• Random Molecular Motion  Molecular heat-flux includes only conduction heat transfer

   Molecular momentum-flux includes both pressure and viscous forces (fluid statics/dynamics)

FORCED CONVECTION

• Advective Transport

   Bulk or macroscopic motion of the fluid

 Such motion, in the presence of a temperature

and velocity gradient contributes to heat and momentum transfer, respectively

ENERGY BALANCE

    

  Unsteady _term ^Conduction _{term = + Convection term}

• _{+ Source term}

  ρ ρ

   ₂ ² ₂ ² ₂ ²

   

   

    

     

   

    

     

   

  Rate of _{Energy flow into CV Rate of Energy flow} out of CV ^{= + Net Viscous Work done} _{on CV}

• ^{+ Rate of accumulation of Energy in} _CV

  Q y T y

    

     

   

   

   

   

    

    

   

  T c _p

  T v x T u p c t

  T x T k z T w y

    

VISCOUS WORK

  Work done by surface stresses in x-direction

ENERGY EQUATION

  S _E = PE +

^q

  Q If work done by surface stresses are included KINETIC AND INTERNAL ENERGY

ENTHALPY EQUATION

• An alternative form of the energy equation is the total enthalpy equation.

   Specific enthalpy h = i + p/ρ _{2 2 2} = h + ½ (u +v +w ) = E + p/ ρ

  Total enthalpy h

TRANSPORT EQUATIONS

   = pressure, three velocity components, enthalpy, temperature and density

  CONSERVATION EQUATION _{and associated scalar fields.}

solved by finite volume based CFD programs to calculate the flow pattern

  1-dim, STEADY CONVECTION-DIFFUSION U _o

  T _o ^{ dx dT} _{ dx dT} ^{U o} _T

^{T- T}

_{1 > T o} ^{w Slug Flow through a long channel of length L}

  2 ² x T x T

  U _o  

    

  α ₂ Pe  

   θ θ

  α θ ^{L U Pe} _L ^{x X} _{T T} ^{T T o} _{o i} ^o   

    ; ;

EXACT SOLUTION

  Pe 

  0 and large diffusivity,

• For small U _o

   the solution is T= x (T linear in x )

  Pe>>0

• For large U o

   Φ grows slowly with x and them suddenly rises to Φ _L ^{over a short} _{distance close to x=L}

  1 ) exp( 1 ) exp( ) ( 

    Pe Pex

  T x

NUMERICAL SOLUTION

  In terms of Central difference Scheme (CDS)    _{2 Pe} θ θ θ θ θ _{i     1 i 1 i 1 i i 1}

   _{2 1 2 ( )}  ^{X  X} Where Pe = Pe Δx   Pe   Pe c _{2 2} θ    θ   θ  _{i c i  4 1 c i  1 •} For Pe ≥ 2 _{Solution shows wiggles, a computational instability c •} oscillation which disappears on fine grids as Pe becomes less than 2 _c

  If upstream differencing is used in convection term?

UPWIND SCHEME

    ^{2 } _{Pe } θ θ θ θ θ _{i i 2  (  ) X    1 i 1 i i X 2 1 Known as First Order Upwind scheme 1  } Pe    θ θ  _{c i   1 i 1}

   θ _{i 2  Pe c For large Pe diffusion term is overestimated} Solution is stable _c

  Hence refined grid is selected at large Pe

_c

OTHER SCHEMES

• Quest for an optimal (stable as well as accurate)

  advection discretization procedure continues

• Other promising alternatives are higher order

  schemes such as Second Order Upwind and QUICK

• QUICK has been found better as compared to the

  SOU advection scheme

  FLOW IN PARALLEL PLATES _{T m 1. Constant heat flux Assumptions:} T  T _{s 3. Steady Flow}

  2. Thermally fully developed flow h

    ucTdy T ( x ) T ( y , x )

    _s ρ

   s    T _m   x T ( x )  T ( y , x ) _{s m}   m c

  

  At the plate surface ) ( ) ( ) ( ) ( ) (

x f

T x x T y T k T x x T q _{m s m s s}

     

     ) (x f h _c  ^{Convective heat transfer Coefficient}

  FLOW IN PARALLEL PLATES

  Energy balance on CV dx x

  T c m x q ^m  

     

  2 FLOW IN PARALLEL PLATES FLOW IN PARALLEL PLATES

  2 ²

     

     T  T  T  T  T

   

  c  c u  v  k 

   

  ρ ρ _p

   

  p ^{2 2}

     

   t  x  y  x  y  

     

   

  2 ^{2 2}

          T   T  T  T

   

  c u   k

    

  ρ _{2 2 2} p

         

   x  y  x  y  

    ₂       _Solving  

     T  T

   

  h D _{c h} u 

   

  α ₂

   8 .

  24  

     x  y

   

  k

      _{D =2R h} CIRCULAR POISEUILLE FLOW

  OUTLINE

• Introduction • Conduction • Convection

   Forced Convection Free Convection

  

• Other Models • Conclusion

NATURAL CONVECTION

  Hot Surface Upward _{Cold Surface Downward}

• ^{Hot Surface Downward Cold Surface Upward}

NATURAL CONVECTION

• The fluid motion is induced by the heat transfer
• Density and temperature are related; hotter gases

  rise…

• Thermal expansion coefficient is a characteristic

  property of fluids

• The momentum equation must be written in

  compressible form (density is varying) and includes a source term

FREE CONVECTION

  α  

_

  At y = 0 ^    T T v u ; ^{At τ= 0} _{For τ> 0 }    ^{T T v u ;} _{s T T v u    ;} ^{y,v x,u}

  Flow over a heated vertical flat plate Boundary conditions: ^{Energy equation Momentum equation Continuity equation} _{At x = 0}

      y v x u

  β ν ₂ ²  

      _{T T g y} ^u _y ^u _{v x} ^u _{u t} ^u

     

      

    

    

   

   

    

     

   

    

     

   

   ₂ ^{T 2} _y ^T _{y v x} ^T _{u t} ^T

FREE CONVECTION

• • The momentum and energy equations are coupled by

  via the temperature Typically called Boussinesq fluids…

• Boussinesq Model: Model treats density as a

  

constant value in all solved equations, except for the

buoyancy term in the momentum equation
• the Boussinesq approximation is valid when

  (T − T ) << 1 β

  OUTLINE

• Introduction • Conduction • Convection

   Forced Convection  Free Convection

• Other Models
• Conclusion
Conjugate Heat Transfer

• • Ability to compute conduction of heat through solids,

  coupled with convective heat transfer in fluid

• In 2D Cartesian coordinates:

  W = wall

   

    

    

     

    

  

  ∂ ∂ ∂ ∂

  ∂ ∂ ∂ ∂

  ρ ∂ ∂

• Properties varies with location and Temperature  
^{T q} _y ^k _{y x}

^{T k} _x ^{T c} _{t w w w w} Periodic Heat Transfer

• Used when flow and heat transfer patterns are repeated  Compact heat exchangers  Flow across tube banks
• Outflow at one periodic boundary is inflow at the other
• Geometry and boundary conditions repeat in streamwise

  direction inflow _outflow

  OUTLINE

• Introduction • Conduction • Convection

   Forced Convection  Free Convection

• Other Models
• Conclusion

  OPTIMIZATION

• Increasing the area A, e.g. by using profiled pipes and ribbed surfaces.
• Increasing ΔT (which is not always controllable).
• For conduction, increasing kf /d.
• Increase h by not relying on natural

  convection, but introducing forced convection

  CONCLUSION

• Thermal conditions at walls, flow boundaries and fluid properties required for energy equation.
• • Chemical reactions, such as combustion, can lead to source terms to be included in the enthalpy equation.
• • Analytical solutions exist for some simple problems and we must rely on computational methods to solve most industrially relevant applications.

THANK YOU

• • An energy equation must be solved together with