Xdc_part2

Still One-Dimensional but More Complicated Heat Transfer from Extended Surfaces Heat flow Heat flux: Heat flow:

q′x′ = h ⋅ (Ts − T∞ ) Q′′ = h ⋅ A ⋅ (Ts − T∞ ) Heat transfer surface area

Fluid flow

Heat flow

Th

Extended surfaces

Tb (base temperature) Ts (surface temperature) T∞ Q′′ = h ⋅ A1 ⋅ (Tb − T∞ ) + h ⋅ A2 ⋅ (Ts − T∞ ) Base area (A1) Fluid flow Extended surface area (A2)

1

T∞ , h

Fin with uniform cross-section Thickness =t Width = w

Conduction

Tb

Ac (cross-section)

Heat flow Length = L

Convection

Tb T(L)

0

x

L

General Analysis dAs qx

dqconv Ac(x)

qf

qx+dx dx z

y x

2

q x = q x + dx + dq x + dx dT dx dq dT d  dT  = q x + x = −k ⋅ Ac ⋅ − k ⋅  Ac ⋅  ⋅ dx dx dx dx  dx 

q x = −k ⋅ Ac ⋅ q x + dx

d  dT   Ac ⋅  ⋅ dx dx  dx  = h ⋅ dAs ⋅ (T − T∞ )

dqconv = −k ⋅ dqconv

d  dT  h dAs ⋅ (T − T∞ ) = 0  Ac ⋅ − ⋅ dx  dx  k dx or d 2T 1 dAc dT 1 h dAs + ⋅ ⋅ − ⋅ ⋅ ⋅ (T − T∞ ) = 0 dx 2 Ac dx dx Ac k dx

Fin Performance Fin effectiveness: The ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin. Heat flow out of the fin

εf =

qf

h ⋅ Ac ,b ⋅ (Tb − T∞ ) Heat flow without the fin

To be practical

εf >2

Fin cross-sectional area at the base

3

Fin efficiency:

ηf =

qf qmax, f

=

qf

h ⋅ A f ⋅ (Tb − T∞ ) Surface area of the fin

Overall surface efficiency:

ηo =

qt qmax,t

=

Total heat flow with the fin

qt h ⋅ At ⋅ (Tb − T∞ ) Total surface area

Total heat flow: qt = h ⋅ Ab ⋅ (Tb − T∞ ) + h ⋅ A f ⋅η f ⋅ (Tb − T∞ ) At = Ab + A f

ηo = 1 −

Af At

⋅ (1 − η f )

4

Transient Heat Transfer (The Uniform Temperature Assumption)

h

T

T∞

T∞

dT mC p ⋅ = h ⋅ A ⋅ (T∞ − T ) dt

∂T ∂ 2T ρC p ⋅ =k 2 ∂t ∂x

or

Biot Number Definition - 1 q”x Ts,1 Ts,2

L T∞ x x=L

x=0

q ′x′ = k ⋅

Ts ,1 − Ts , 2 L

= h ⋅ (Ts , 2 − T∞ )

1

Biot Number Definition - 2 q ′x′ = k ⋅

Ts ,1 − Ts , 2 L Ts ,1 − Ts , 2 Ts , 2 − T∞ Bi =

= h ⋅ (Ts , 2 − T∞ ) =

h⋅L k

h⋅L k

Critical Biot Number: Bi < 0.1

Non-uniform Temperature T

h T∞

T∞

k 1   R h⋅L L Bi = =   = conduction ks 1h Rconvection

2

Typical Values of the Convection Heat Transfer Coefficient (h) Process Free convection Gases Liquids Forced convection Gases Liquids Convection with phase change Boiling or condensation

h (W.m-2.K-1) 2-25 50-1000 25-250 100-20,000

2500-100,000

Determination of h (1) External flow

Double-pipe arrangement

Internal flow

Pipe

Channel

3

Determination of h (2) Forced convection is due to fluid movement which enhances heat transfer. Conduction only Fluid temperature T∞ Convection

m ⋅ C p ⋅T travels at a Velocity Hot object

Determination of h (3) Conduction: Thermal conductivity of the fluid (k) * Thermal diffusivity:  

α = k  (m2.s-1)  ρ ⋅ C p  

Density of the fluid (ρ)

Convection: Velocity and its distribution (u or V) Viscosity of the fluid (µ) * Kinematic viscosity: υ = µ 

 

ρ 

(m2.s-1)

Density of the fluid (ρ) Specific heat capacity of the fluid (Cp)

4

Determination of h (4) (Dimensionless Groups) Heat transfer: Biot number (Bi) Reynolds number (Re) Prandtl number (Pr) Nusselt number (Nu) Fourier number (Fo)

Film temperature :

T film =

Ts + T∞ 2

Mass transfer: Sherwood number (Sh) Schmidt number (Sc)

Determination of h (5) (Construction of the Correlations)

Rate of ‘reaction’

Nu =

h⋅L kf

Reactant A

Re =

ρ ⋅u ⋅ L µ

Reactant B

Pr =

υ α

Nu = c ⋅ Re m ⋅ Pr n

5

The Loop Tbulk fluid Tsurface

Physical properties

Dimensionless parameters Re, Pr etc

Another estimate? Do you really know that?

Nu ⋅ k f

h= L/L kfluid Nu

Find the correlation for Nu

6

Transient Conduction (Finite difference method) One dimensional (at x-direction):

∂T ∂T ρ ⋅Cp ⋅ =k⋅ 2 ∂t ∂x 2

Th

Steady state

At t > 0

Tl

Initial temperature (t = 0) To

Initial and Boundary Conditions Initial condition:

t = 0 ,T ( x ) = To , x ∈ [0 , L ]

Th

Steady state

Boundary condition:

∂T = hh ⋅ (Th − T (t ,0 )) ∂x ∂T t > 0 , x = L ,− k = hl ⋅ (T (t , L ) − Tl ) ∂x t > 0 , x = 0 ,− k

x=0

To

Tl

x=L

1

When heat transfer coefficients are very large (+∞) Th

Steady state

Tl

t ≥ 0 , x = 0,T = Th t ≥ 0 , x = L ,T = Tl

Finite Difference Approach ∂T ∂ 2T =k⋅ 2 ρ ⋅Cp ⋅ ∂t ∂x T p+1 2

3

p

∂T ∆T T p +1 − T p ≈ = ∂t ∆t ∆t

1

t=0

∆t

t

2

T − Ti ∆T  ∂T  = i +1   ≈ ∆x i + 0.5 ∆x  ∂x i + 0.5  p

1

T

2

p

p

p

3 i-1

i

i+1

T − Ti −1 ∆T   ∂T = i  ≈  ∆x i −0.5 ∆x  ∂x i −0.5  p

x=L

∆x

x=0 p

p

p

∆T ∆T p − 2 ∂ T   2  ≈ ∆x i + 0.5 ∆x i −0.5 ∆x  ∂x i  Ti +p1 − Ti p Ti p − Ti −p1 − ∆ ∆x x = ∆x Ti +p1 − 2Ti p + Ti −p1 = (∆x )2 p

Explicit Method p +1

T − Ti ∂T ≈ i ∆t ∂t i p

p

p

3

p +1

T − Ti ∂T ≈ ρC p i ρC p ∂t i ∆t p

Ti

Ti

p +1

p +1

− Ti

p

Ti +p1 − 2Ti p + Ti −p1 k⋅ (∆x )2

p

α ⋅ ∆t

(∆x )

2

(

⋅ Ti +p1 − 2Ti p + Ti −p1

)

 2 ⋅ α ⋅ ∆t  p p p 1 −  ⋅ Ti ⋅ + + T T i +1 i −1 2 2  (∆x ) (∆x )  

α ⋅ ∆t

(

)

4

Other Boundary Conditions Steady state

Th

∂T = hh ⋅ (Th − T (t ,0 )) ∂x ∂T t > 0 , x = L ,−k =0 ∂x t > 0 , x = 0 ,− k

To

Adiabatic

Adiabatic

x=0

x=L

Exact Equation ∂T ∂ 2T =k⋅ 2 ρ ⋅Cp ⋅ ∂t ∂x

Approximate Equation Ti

p +1

 2 ⋅ α ⋅ ∆t  p p p 1 −  ⋅T ⋅ + + T T (∆x )2 i +1 i −1  (∆x )2  i

α ⋅ ∆t

(

)

1

Approximate Equation Ti

 2 ⋅ α ⋅ ∆t  p p p 1 −  ⋅T ⋅ + + T T (∆x )2 i +1 i −1  (∆x )2  i

α ⋅ ∆t

p +1

(

)

Setting this One could then deduce a ∆t once α and ∆x are given

Ti

p +1

α ⋅ ∆t

(∆x )

2

0?

 2 ⋅ α ⋅ ∆t  1 − =0 2  (∆x )  

(

⋅ Ti +p1 + Ti −p1

)

An Example An medium (solid), which has uniform initial temperature of 50°C and a length of 0.5m, has its one side suddenly exposed to a fast flowing heating medium (fluid) having very large h (giving a constant surface temperature Ts = 150°C). The density of the solid is 8933 kg.m-3, specific heat capacity is 385 J.kg-1.K-1, thermal conductivity is 401 W.m-1.K-1. Show the temperature profile as heating progresses.

2

Convective Heat (& Mass) Transfer – Fundamentals 1 (at interface between two mediums) x

Solid

The heat flux along x-direction:

q′x′

=

− ks ⋅

∂T ∂T = −k f ⋅ ∂x s ∂x

Ts

h Fluid

f

T∞

∂T = h ⋅ (Ts − T∞ ) − ks ⋅ ∂x s

kf ⋅ h=−

∂T ∂x

f

(Ts − T∞ )

Average heat transfer coefficient

q′′

u∞, T∞

dAs As, Ts ‘Total’ heat flow (W.m-2): Local heat flux (W.m-2):

q′′ = h ⋅ (Ts − T∞

q )

Local heat transfer coef (W.m-2.K-1) Average heat transfer coef (W.m-2.K-1)

= ∫ h ⋅ (Ts − T∞ ) ⋅ dAs As

Constant surf temp

   h ⋅ dA  s  A∫  s   ⋅ (T − T ) ⋅ A ≈ s ∞ s As

1

Average heat transfer coefficient (flat plate)

q′′

u∞, T∞

As, Ts x

dx Plate length L Average heat transfer coef (W.m-2.K-1):

‘Total’ heat flow (W.m-2):

q

L

= h ⋅ L ⋅ (Ts − T∞ )

h=

∫ h ⋅ dx 0

L

Mass transfer (flat plate)

N ′′

u∞, cA,∞

(local mass flux, kg.m-3.m-2.s-1)

As, cA,s x

dx Plate length L Average mass transfer coef (m.s-1):

N ′A′ = hm ⋅ (c A, s − c A,∞ )

N A = hm ⋅ As ⋅ (c A, s − c A,∞ ) hm =

L

∫h

m

⋅ dx

0

L

2

Convection Boundary Layers (Velocity and temperature) Velocity Profile Temperature Profile Temperature

T∞

u∞

y

Velocity

δT

Heat Flux

δV

Ts

x Solid or Liquid Surface

Heated Plate

Convection Boundary Layers (Velocity and concentration) Velocity Profile Concentration Profile Concentration

cA,∞

u∞

y

Velocity

Mass Flux

δV

δC cA,s

x Solid or Liquid Surface

3

Heat and mass transfer coefficients kf ⋅ h=−

∂T ∂y

DA ⋅ y =0

hm = −

(Ts − T∞ )

(c

A, s

∂c A ∂y

y =0

− c A ,∞ )

y

h or hm

x Stationary medium

Laminar to Turbulent Flow (Transition) Velocity Profile Velocity Profile

u∞

y

Re x ,c = −

ρ ⋅ u∞ ⋅ xc = 5 ×105 µ

u∞

Turbulent region Buffer layer

δV

x

xc Laminar

Turbulent

Laminar sublayer

Transition Solid or Liquid Surface

4

Convective Heat (& Mass) Transfer – Fundamentals 1 (at interface between two mediums) x

Solid

The heat flux along x-direction:

q′x′

=

− ks ⋅

∂T ∂T = −k f ⋅ ∂x s ∂x

Ts

h Fluid

f

T∞

∂T = h ⋅ (Ts − T∞ ) − ks ⋅ ∂x s

kf ⋅ h=−

∂T ∂x

f

(Ts − T∞ )

Average heat transfer coefficient (flat plate)

q′′

u∞, T∞

As, Ts x

dx Plate length L

‘Total’ heat flow (W.m-2):

q

Average heat transfer coef (W.m-2.K-1):

L

= h ⋅ L ⋅ (Ts − T∞ )

h=

∫ h ⋅ dx 0

L

1

Mass transfer (flat plate)

N ′′

u∞, cA,∞

(local mass flux, kg.m-3.m-2.s-1)

As, cA,s x

dx Plate length L Average mass transfer coef (m.s-1):

N ′A′ = hm ⋅ (c A, s − c A,∞ )

L

N A = hm ⋅ As ⋅ (c A, s − c A,∞ ) hm =

∫h

m

⋅ dx

0

L

Heat and mass transfer coefficients kf ⋅ h=−

∂T ∂y

DA ⋅ y =0

(Ts − T∞ )

hm = −

(c

A, s

∂c A ∂y

y =0

− c A ,∞ )

y

h or hm

x Stationary medium

2

Laminar to Turbulent Flow (Transition) Velocity Profile Velocity Profile

Re x ,c =

u∞

y

ρ ⋅ u∞ ⋅ xc = 5 ×105 µ

u∞

Turbulent region Buffer layer

δV

x

xc Laminar

Turbulent

Laminar sublayer

Transition Solid or Liquid Surface

Convective Heat (& Mass) Transfer – Fundamentals 2 (h and hm determination) Conduction (heat): Thermal conductivity of the fluid (k) *Thermal diffusivity (α) (m2.s-1) Density of the fluid (ρ) Temperature gradient Diffusion (mass): *Mass diffusivity (D) (m2.s-1) Density of the fluid (ρ) Concentration gradient Convection: Velocity and its distribution (u or V) Viscosity of the fluid (µ) * Kinematic viscosity (υ) (m2.s-1) Density of the fluid (ρ) Specific heat capacity of the fluid (Cp)

h(W .m −2 .K −1 )

hm (m.s −1 )

Temperature and concentration difference

3

Determination of h Heat transfer: Biot number (Bi) Reynolds number (Re) Prandtl number (Pr) Nusselt number (Nu) Fourier number (Fo)

Film temperature :

T film =

Ts + T∞ 2

Film concentration :

Mass transfer: Sherwood number (Sh) Schmidt number (Sc)

c A, film =

c A, s + c A, ∞ 2

Determination of h Rate of ‘reaction’

Nu =

h⋅L kf

Reactant A

Re =

ρ ⋅u ⋅ L µ

Reactant B

Pr =

υ α

Nu = c ⋅ Re m ⋅ Pr n Sh = c ⋅ Re m ⋅ Sc n

4

The Working Loop Tbulk fluid Tsurface

Physical properties

Dimensionless parameters Re, Pr etc

Another estimate? Do you really know that?

Nu ⋅ k f

h= L/L kfluid Nu

Find the correlation for Nu

5

Convective Heat (& Mass) Transfer – Fundamentals 3 (Internal flows - momentum)

Re =

Wall Uniform Entry Flow Velocity u∞

δv

Fully developed region

xfd,h Fully developed

r

ro

Entry flow region

x

ρ ⋅ um ⋅ d µ

Hydrodynamic

Laminar to Turbulence Transition Re c ≈ 2300

Flow Becomes Fully Turbulent Re > 4000

Laminar Entry Length x fd ,h d

≈ 0.05 Re

Hydraulic Diameter

dh =

4 Ac P

Turbulent Entry Length 10 ≤

x fd ,h d

≤ 60

1

(Internal flows - energy)

Re =

ρ ⋅ um ⋅ d µ

Temperature profile Wall temperature Uniform Entry Temperature Tin

δT

ro

Thermal entry region

r

Fully developed region

xfd,t x Fully developed

Thermal

Laminar Thermal Entry Length x fd ,t ≈ 0.05 Re⋅ Pr

Turbulent Entry Length x fd ,t d

≈ 10

Fully developed thermal region

Comparison When Pr > 1,

x fd ,h < x fd ,t

When Pr < 1,

x fd ,h > x fd ,t

When Pr >100,

x fd ,h << x fd ,t

∂  Ts ( x ) − T (r , x )  =0  ∂x  Ts ( x ) − Tm ( x ) 

2