Heat Transfer Handbook part 3 potx

The low-velocity, or laminar ﬂow, average convective heat transfer coefﬁcient along a surface of length L for Re ≤ 3 × 105is found to be h = 0.664 k LRe1/2· Pr1/3 W/m2· K 1.17 wherek is

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d e≡4A

p

In the inlet zones of such parallel-plate channels and along isolated plates, the heat transfer coefﬁcient varies with the distance from the leading edge The low-velocity,

or laminar ﬂow, average convective heat transfer coefﬁcient along a surface of length

L for Re ≤ 3 × 105is found to be

h = 0.664 k

LRe1/2· Pr1/3 (W/m2· K) (1.17)

wherek is the ﬂuid thermal conductivity, L the characteristic dimension of the

sur-face, and Re the Reynolds number based on L: namely, ˆV L/ν.

A similar relation applies to a ﬂow in tubes, pipes, annuli or channels, with the equivalent diameterd e serving as the characteristic dimension in both the Nusselt and Reynolds numbers For laminar ﬂow, Re≤ 2100,

hd e

k = 1.86

Re· Prd e

L

1/3 µ

µw

0.14

(1.18)

which is attributed to Sieder and Tate (1936) and whereµw is the viscosity of the convective medium at the channel wall temperature Observe that this relationship shows that the heat transfer coefﬁcient attains its maximum value at the inlet to the channel and decreases asd e /L decreases.

In higher-velocity turbulent ﬂow along plates, the dependence of the convective heat transfer coefﬁcient on the Reynolds number increases, and in the range Re ≥

3× 105,

h = 0.036 k

LRe0.8· Pr1/3 (W/m2· K) (1.19)

In pipes, tubes, annuli, and channels, turbulent ﬂow occurs at an equivalent diameter-based Reynolds number of 10,000, with the regime bracketed by 2100≤

Re≤ 10,000 usually referred to as the transition region For the transition region,

Hausen (1943) has provided the correlating equation

hd e

k = 0.116(Re − 125)Pr1/3 1+

d e

L

2/3 µ

µw

0.14

(1.20)

and Sieder and Tate (1936) give for turbulent ﬂow

hd e

k = 0.023Re0.8· Pr1/3

µ

µw

0.14

(1.21) Forced convection in internal and external ﬂows is treated in greater detail in Chapters

5 and 6

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Boiling heat transfer displays a complex dependence on the temperature difference between the heated surface and the saturation temperature (boiling point) of the liquid Following Rohsenow (1952), the heat transfer rate in nucleate boiling, the primary region of interest, can be approximated by a relation of the form

qφ= C sf A(T s − Tsat)3 (W) (1.22) whereC sf is a function of the surface–ﬂuid combination For comparison purposes,

it is possible to deﬁne a boiling heat transfer coefﬁcienthφ:

hφ = C sf (T s − Tsat)2 (W/m2· K)

which, however, will vary strongly with surface temperature Boiling and condensa-tion are treated in greater detail in Chapters 9 and 10, respectively

Frequent use is made of ﬁnned or extended surfaces, and while such ﬁnning can substantially increase the surface area in contact with the coolant, conduction in the

fin reduces the average temperature of the exposed surface relative to the fin base In the analysis of such finned surfaces, it is common to define a fin efficiencyη as being

equal to the ratio of the actual heat dissipated by the fin to the heat that would be dissipated if the fin possessed an infinite thermal conductivity Using this approach, heat transferred from a fin or a fin structure can be expressed in the form

q f = hS f η(T b − T s ) (W) (1.23) whereS f is the surface area of the ﬁn,T b the temperature at the base of the ﬁn,T s

the surrounding temperature, andq b the heat entering the base of the fin, which in the steady state is equal to the heat dissipated by the fin The thermal resistance of a finned surface is given by

R f ≡ 1

andη is approximately 0.63 for a thermally optimum rectangular-cross-section ﬁn

The transfer of heat to a ﬂowing gas or liquid that is not undergoing a phase change results in an increase in the coolant temperature from an inlet temperature ofTinto

an outlet temperature ofTout, according to

q = ˙mc p (Tout− Tin) (W) (1.25)

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Based on this relation, it is possible to deﬁne an effective ﬂow resistance,R f l, as

R f l≡ 1

Unlike conduction and convection, radiative heat transfer between two surfaces or between a surface and its surroundings is not linearly dependent on the temperature difference and is expressed instead as

q = σA F

T4

1 − T4 2

whereF includes the effects of surface properties and geometry andσ is the Stefan–

Boltzmann constant,σ = 5.669 × 10−8W/m2· K4 For modest temperature differ-ences, this equation can be linearized to the form

q r = h r S(T1− T2) (W) (1.28) whereh ris the effective “radiation” heat transfer coefﬁcient,

h r = σFT2

1 + T2 2

(T1+ T2) (W/m2· K) (1.29a) and for small∆T = T1− T2is approximately equal to

h r = 4σF(T1T2)3/2 (W/m2· K) (1.29b)

It is of interest to note that for temperature differences on the order of 10 K, the radiative heat transfer coefﬁcienth rfor an ideal (or “black”) surface in an absorbing environment is approximately equal to the heat transfer coefﬁcient in natural convec-tion of air Noting the form of eq (1.27), the radiaconvec-tion thermal resistance, analogous

to the convective resistance, is seen to equal

R r ≡ 1

Heat transfer and fluid flow analyses of objects of various sizes and shapes and their corresponding flow fields are facilitated by working in a coordinate system that provides a good fit to the flow geometry Figure 1.5 presents diagrams for the rectangular (Cartesian), cylindrical, and spherical coordinate systems Equations for the gradient of a scalar, divergence and curl of a vector, and the Laplacian are given below for the three coordinate systems

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r

␪

␪ = 0

z = 0

␪ =0

␾

r

y

ˆV y

ˆV x

ˆV z

ˆV r

ˆVθ

ˆVθ ˆVφ

x z

Origin

Figure 1.5 Rectangular (Cartesian), cylindrical, and spherical coordinate systems

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For a rectangular coordinate system with coordinatesx,y, and z and unit vectors e x,

ey, and ez, the gradient of the scalarT is

gradT = ∇T = e x

∂T

∂x + ey

∂T

∂y + ez

∂T

The divergence of a vector V having componentsV x , V y, andV zis

div V = ∇ · V = ∂V ∂x x +∂V ∂y y +∂V ∂z z (1.32)

The curl of the vector V is

curl V = ∇ × V = ex

∂V

z

∂y −

∂V y

∂z

+ ey

∂V x

∂z −

∂V z

∂x

+ ez

∂V y

∂x −

∂V x

∂y

(1.33)

Alternatively, curl V may be written as the determinant

∂

∂x

∂

∂y

∂

∂z

V x V y V z

(1.34)

The Laplacian of the scalarT is

∇2T = ∂2T

∂x2 +∂2T

∂y2 +∂2T

For a cylindrical coordinate system with coordinatesr, θ, and z and unit vectors e r

eθ, and ez, the gradient of the scalarT is

gradT = ∇T = e r ∂T

∂r + eθ

1

r

∂T

∂θ + ez

∂T

The divergence of a vector V having components,V r , Vθ, andV zis

div V = ∇ · V = 1r ∂(rV ∂r r )+1r ∂V ∂θθ +∂V ∂z z (1.37)

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The curl of the vector V is

curl V = ∇ × V = er

1

r

∂V z

∂θ −

∂Vθ

∂z

+ eθ

∂V r

∂z −

∂V z

∂r

+ ez

1

r

∂(rVθ)

∂r −

∂V r

∂θ

(1.38)

Alternatively, curl V may be written as the determinant

er1

1

r

∂

∂r

∂

∂θ

∂

∂z

V r rVθ V z

(1.39)

∇2T = 1

r

∂

∂r

r∂T

∂r

+ 1

r2

∂2T

∂θ2 +∂2T

For a spherical coordinate system with coordinatesr, θ, and φ and unit vectors e r, eθ,

and eφ, the gradient of the scalarT is

gradT = ∇T = e r ∂T

∂r + eφ

1

r

∂T

∂φ + eθ

1

r sin φ

∂T

The divergence of a vector V having components,V r Vθ, andVφis

div V = ∇ · V = 1

r2

∂(r2V r )

1

r sin φ

∂(Vφsinφ)

1

r sin φ

∂Vθ

∂θ (1.42)

The curl of the vector V is curl V = ∇ × V = er r sin φ1

∂ (Vθsinφ)

∂Vφ

∂θ

+ eφ1r

1 sinφ

∂V r

∂θ −

∂(rVθ)

∂r

+ eθ1r ∂

rVφ

∂V r

∂φ

(1.43)

∇2T = r12∂r ∂

r2∂T

∂r

+r2 1

sinφ

∂ dφ

sinφ∂T ∂φ

r2sin2φ

∂2T

∂θ2 (1.44)

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In general, a curvilinear coordinate system can be proposed where a vector V has

componentsV1, V2, andV3in thex1, x2, andx3coordinate directions The unit vectors

are e1, e2, and e3in the coordinate directionsx1, x2, andx3and there are scale factors,

s1, s2, ands3that relate the general curvilinear coordinate system to the rectangular, cylindrical, and spherical coordinate systems

In the general curvilinear coordinate system, the gradient of a scalarT is

gradT = ∇T = e1

1

s1

∂T

∂x1

+ e2

1

s2

∂T

∂x2

+ e3

1

s3

∂T

∂x3

(1.45)

The divergence of a vector V having components,V1, V2, andV3is

div V = ∇ · V = 1

s1s2s3

∂(s2s3V1)

∂x1

+∂(s3s1V2)

∂x2

+∂(s1s2V3)

∂x3

(1.46)

The curl of the vector V is curl V = ∇ × V = e1

1

s2s3

∂(s3V3)

∂x2

−∂(s2V2)

∂x3

+ e2

1

s1s3

∂(s

1V1)

∂x3

−∂(s3V3)

∂x1

+ e3

1

s1s2

∂(s

2V2)

∂x1

−∂(s1V1)

∂x2

(1.47) The Laplacian of the scalarT is

∇2T = 1

s1s2s3

∂

∂x1

s2s3

s1

∂T

∂x1

+∂x ∂ 2

s

1s3

s2

∂T

∂x2

+∂x ∂ 3

s

1s2

s3

∂T

∂x3

(1.48)

In eqs (1.45) through (1.48), the conversion from the general curvilinear coordi-nate system to the rectangular, cylindrical, and spherical coordicoordi-nate systems depends

on the assignment of values to the coordinatesx1, x2, and x3 and the scale factors

s1, s2, ands3

x1 x2 x3 s1 s2 s3

A control volume is a region in space selected for analysis An incremental control volume carrying a mass ﬂux is shown in Fig 1.6 The conservation of mass principle

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can be applied to the control volume by noting that the net rate of mass ﬂux out of the control volume plus the rate of accumulation of mass within the control volume must equal zero:

A

ρˆV · ndA = ∂

∂t

V

Observe that the mass within the control volume is

m = ρ ∆x ∆y ∆z

and that the mass ﬂux at each of the faces of Fig 1.6 will beρ( ˆV · n), where n is the

normal to the areadA.

Noting that the density can vary from point to point and with time,ρ = f (x,y,z,

t), the net mass ﬂux out of the control volumes in each of the coordinate directions

will be

ρ ˆV x|x+∆x − ρ ˆV x|x∆y ∆z

ρ ˆV y|y+∆y − ρ ˆV y|y

∆x ∆z

ρ ˆV z|z+∆z − ρ ˆV z|z

∆x ∆y

With all of the foregoing substituted into eq (1.49),

ρ ˆV x|x∆x − ρ ˆV x|x∆y ∆z +ρ ˆV y|y+∆y − ρ ˆV y|y∆x ∆z

+ρ ˆV z|z+∆z − ρ ˆV z|z

∆x ∆y + ∂ρ ∂t ∆x ∆y ∆z = 0

Figure 1.6 Mass ﬂux through an incremental ﬂow volume

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and after division by∆z ∆y ∆z, the result is ρ∂ ˆV x|x+∆x − ρ ˆV x|x

ρ ˆV y|y+∆y − ρ ˆV y|y

ρ ˆV z|z+∆z − ρ ˆV z|z

∂ρ

∂t = 0

In the limit as∆x, ∆y, and ∆z all tend to zero, the result is

∂

∂x (ρ ˆV x ) +

∂

∂y (ρ ˆV y ) +

∂

∂z (ρ ˆV z ) +

∂ρ

This may be written as

∇ · ρ ˆV +∂ρ

where∇ · ρ ˆV = div ρ ˆV is the divergence of the vector ˆV This equation is general: It

applies to unsteady three-dimensional ﬂow with variableρ

Equation (1.51) is a vector equation that represents the equation of continuity in

rectangular, cylindrical, and spherical coordinates If the ﬂow is incompressible, so thatρ is independent of time, eq (1.51) reduces to

which applies to both steady and unsteady ﬂow This equation is also a vector equation that applies to rectangular, cylindrical, and spherical coordinates

Equation (1.50) can be written as

∂ρ

∂t + ˆV x

∂ρ

∂x + ˆV y

∂ρ

∂y + ˆV z

∂ρ

∂z + ρ

∂ ˆV x

∂x +

∂ ˆV y

∂y +

∂ ˆV z

∂z

= 0

or

Dρ

where

Dρ

Dt =

∂ρ

∂t + ˆV x

∂ρ

∂x + ˆV y

∂ρ

∂y + ˆV z

∂ρ

is called the substantial derivative of the densityρ Thus, in rectangular coordinates,

D

Dt =

∂

∂t + ˆV x

∂

∂x + ˆV y

∂

∂y + ˆV z

∂

The substantial derivative in cylindrical coordinates is

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D

Dt =

∂

∂t + ˆV r

∂

∂r +

ˆVθ

r

∂

∂θ + ˆV z

∂

and in spherical coordinates, it is

D

Dt =

∂

∂t + ˆV r

∂

∂r +

ˆVφ

r

∂

∂φ+

ˆVθ

r sin φ

∂

The equation of continuity representing the conservation of mass can be summa-rized for the three coordinate systems For the rectangular coordinate system,

∂ρ

∂t +

∂(ρ ˆV x )

∂x +

∂(ρ ˆV y )

∂(ρ ˆV z )

for the cylindrical coordinate system,

∂ρ

∂t +

1

r

∂(ρr ˆV r )

1

r

∂(ρ ˆVθ)

∂θ +

∂(ρ ˆV z )

and for the spherical coordinate system,

∂ρ

∂t +

1

r2

∂(ρr2ˆV r )

1

r sin φ

∂(ρ ˆVφ sin φ)

1

r sin φ

∂(ρ ˆVθ)

∂θ = 0 (1.60)

In the event that the ﬂow may be modeled as incompressible,

∂ ˆV x

∂x +

∂ ˆV y

∂y +

∂ ˆV z

for the cylindrical coordinate system,

1

r

∂(r ˆV r )

∂r +

1

r

∂ ˆVθ

∂θ +

∂ ˆV z

and for the spherical coordinate system,

1

r2

∂(r2ˆV r )

1

r sin φ ∂( ˆVφ sin φ)

1

r sin φ

∂ ˆVθ

∂θ = 0 (1.63)

The momentum theorem of fluid mechanics provides a relation between a group of field points It is especially useful when the details of the flow field are more than moderately complicated and it is based on Newton’s law, which can be written as

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