heat transfer collection of formulas and tables_ 2009

HEAT TRANSFER

Collection of formulas and Tables of Thermal Properties Eric Granryd Division of Applied Thermodynamics and Refrigeration Dept. of Energy Technology The Royal Institute of Technology Stockholm 2009

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Distribution: Division of Applied Thermodynamics and Refrigeration Dept. of Energy Technology The Royal Institute of Technology 10044 Stockholm, Sweden + Phone:+46 8 790 7451 Fax: +46 8 20 41 61 The authors have compiled this information with care but cannot guarantee that it is free of errors. The entire risk of use of any information in this publication is assumed by the used. Adopted for use with the textbook “Heat Transfer”, 7th ed. By J.P. Holman by Eric Granryd and Björn Palm Edition 2009 in new layout. Björn Palm has assisted with important viewpoints and valuable contributions.

First edition 1991. Editions with minor changes and additions have appeared 1992, 1995, 1997, 2001, 2003, 2005. © The author and The Division of Applied Thermodynamics and Refrigeration Dept. of Energy Technology The Royal Institute of Technology 10044 Stockholm, Sweden

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CONTENT Collection Of formulas

General 5

Heat exchangers 6

Heat conduction 8

Laws of similarity 12

Forced convection without change of phase -- Pressure drop and heat transfer 14

Heat transfer in free convection 19

Heat transfer in falling films 22

Heat transfer in condensation 24

Heat transfer in boiling liquids 28

Heat transfer in radiation 31

Appendix 35 Heat Exchanger Effectiveness 37-39

Fin Efficiency 40

Shape factors for heat conduction 41-43

Temperatures in transient conduction 44-45

Heat transfer in falling films 46

Heat transfer in nuclear boiling 47

View factors for radiation 48-50

Tables of thermal and physical properties for a selection of solid materials, liquids and gases 51

Table 1. Thermal properties of a selection of metals 52-53

Table 2. Thermal properties of a selection of non metallic materials 54

Table 3. Thermal properties of a selection of building and related materials 55-56

Table 4. Thermal properties of air (Atmospheric air and Humid air) 57-58

Table 5. Thermal properties of water (liquid, solid and vapor) 59

Table 6. Thermal properties of a selection of liquids 60-62

Table 7. Thermal properties of a selection of gases 63-64

Table 8. Thermal properties of a selection of Refrigerants 65-68

Table 9. Physical properties for selected substances (gases) 69-71

Table 10 Heat resistance due to fouling in heat exchangers 73

Units and conversion factors 74-75

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Collection of formulas in

H E A T T R A N S F E R GENERAL Heat Conduction The basic law of heat conduction in a solid is Fouriers law:

xTAkq

∂∂

⋅⋅−= 1-11

where q = heat flow rate (or heat transfer), [W] k = thermal conductivity of the material [W/(m.K) or W/(m.°C)] A = area [m2] T = temperature [K] (or, if temperature is denoted t [°C]) x = distance [m]

Heat transfer by convection “Newton’s law” postulates that the heat transferred from a surface to a fluid is: )( ttAhq w −⋅⋅= 1-8 where h = coefficient of heat transfer [W/(m2.°C)] tw = temperature of wall surface and t = fluid temperature [°C]

Overall heat transfer from fluid through a wall to another fluid in stationary conditions: tAUq Δ⋅⋅= 2-13 where U = overall heat transfer coefficient [W/(m2.°C)] ∆t = temperature difference between fluids on the two sides of the wall [°C]

An expression for the overall heat transfer coefficient, U (or U .A), can be achieved by observing that the ∆t in equation 2-13 is the sum of all temperature differences fluid1 to surface A1, through the wall, and from surface A2 to fluid2. Since q is the same it can be eliminated and the result is the following relation

∑ ⋅+

⋅Δ

+⋅

=⋅ 2211

111AhAk

xAhAU mww

w (Equivalent to 2-15)

where h1 and h2 = coefficient of heat transfer on sides ‘1’ and ‘2’ A1 and A2 = wall area on sides ‘1’ and ‘2’ ∑ ⋅

Δmww

w

Akx = heat transfer resistance in all layers of the wall.

The terms in this equation can be regarded as heat transfer resistances. Each term is proportional to the temperature difference over the layer in question.

In a wall equipped with fins (fin area = Afin) the fin surface temperature will vary due to the conduction resistance in the fins. This can be expressed as if only a fraction fη (= fin efficiency) of the temperature difference is effective. If, for instance side ‘2’ consist of a base wall surface A2w without fins and in addition also equipped with a finned surface Afin, then instead of h2

.A2, use: h2.(A2w+Afin

. fη ) in the equation.

As part of wall resistance also fouling should be included. Examples of tube side resistance due to fouling are given in Appendix, Table 10.

1 Numbers of equations as well as denominations of diagrams and tables given in the text refer to the book ‘Heat Transfer’ by J.P. Holman. References are also given to information in Appendix and Tables.

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HEAT EXCHANGERS The purpose of a heat exchanger is normally to exchange thermal energy from one (hot) fluid to another. In the heat exchanger the temperature of the fluids are changing and a temperature profile is established. There are different arrangements for the fluid flow. In a counter flow heat exchanger the fluids move in opposite directions. Other arrangements are parallel or different types of cross flow. Figures in appendix on page 35 – 37 illustrate profiles and other characteristics. Special cases occur in heat exchangers with phase changing media as in evaporators and condensers where the phase change occurs at a constant temperature.

1. For cases where the inlet and outlet temperatures are known for both fluids the following approach is the simplest:

The capacity of a heat exchanger is proportional to the over-all total heat transfer coefficient of the unit, AU ⋅ , and an average temperature difference, mϑ , between the fluids. One can write:

mAUq ϑ⋅⋅= 10-5 For cases of counter flow and parallel flow heat exchangers (for cases with constant fluid heat capacities and constant U-value) the correct average temperature difference to be used in equation 10-5 is equal to the logarithmic mean temperature difference, mϑ , which can be calculated from the relation:

)/ln( 21

21

ϑϑϑϑ

ϑ−

=m 10-12

where 1ϑ and 2ϑ are the temperature differences at inlet and outlet sides of the heat exchanger (see figures on pages 35-36 in Appendix).

Notice that this relation is not valid for cross flow heat exchangers. A variety of different flow arrangements are possible. For such cases a correction factor “F” can be introduced in equation 10-5. The value of this correction factor can be found by using diagrams, see for instance figures 10-8 – 10-11 in the textbook by Holman. The transferred heat rate, q, will result in a temperature change of the fluids, as defined by the following relation (in which index “h” and “c” refers to the different fluids), expressing a first law energy balance (in the figures in the Appendix m . cp is denoted W ):

cphp tcmtcmq )()( Δ⋅⋅=Δ⋅⋅= && [W] 10-6 2. For cases where the outlet fluid temperatures are to be estimated, the following approach, the so-called NTU-method, is recommended:

The heat exchanger effectiveness, ε, is defined as the ratio of the temperature change, Δt1, of the fluid with the smallest heat capacity and thus largest temperature change divided by the temperature difference between entering fluids, θ. Temperature profiles are illustrated in Appendix, pages 35 – 37. The heat exchanger effectiveness is thus defined as:

7

θ

ε 11

tΔ= 10-21

Define the following symbols:

NTUcmAUXp

=⋅⋅

=1)( &

; and: 2

1

2

1

)()(

WW

cmcm

Yp

p

&

&

&

&=

⋅

⋅=

Chose sides so that Y is < 1. (Side “1” is thus the one with the smallest heat capacity flow.) For counter flow heat exchangers (assuming fluids with constant specific heat capacity and constant U-value) the following relations apply:

)]1(exp[1

)]1(exp[1YXY

YX−⋅−⋅−

−⋅−−=ε 10-27

The temperature differences at sections of the heat exchanger “a” and “b” can be calculated by the following relation: )]1(exp[/ YXba −⋅−=ϑϑ For parallel flow heat exchangers (with fluids with constant specific heat capacity and constant U-value) the following relations apply:

Y

YX+

+⋅−−=

1)]1(exp[1ε 10-26

The temperature differences at sections “a” and “b” can be calculated by the following relation: )]1(exp[/ YXba +⋅−=ϑϑ In Appendix, pages 35 – 37 the heat exchanger effectiveness for counter flow, parallel flow and pure cross flow heat exchangers are given in the form of diagrams (here the symbol η is used instead of ε for the heat exchanger effectiveness). Equivalent figures are Fig. 10-12 – 10-15 in the textbook by Holman. For cases (like evaporators and condensers) where one of the media (side “2”) has a constant or nearly constant temperature (hence Y → 0) the cases of parallel, counter, or cross flow heat exchange give identical results. For this case the result becomes:

)exp(1 X−−=ε The heat rate transferred for cases where the temperature difference between the incoming media, θ, is known and where we have estimated, ε, is simply estimated by:

θε ⋅⋅⋅= 11)( pcmq & [W]

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HEAT CONDUCTION Basic relations a) The Fourier law (as already introduced, page 5):

xTAkq

∂∂

⋅⋅−= 1-1

where q = heat flow rate [W] k = thermal conductivity of the material [W/(m.K) or W/(m.°C) ] A = area [m2] T = temperature [K] (or, if temperature is denoted t [°C]) x = distance [m] b) The Fourier equation (in the form written here it is valid only for isotropic media with

temperature independent thermal conductivity, density and specific heat capacity)

pc

qzT

yT

xTT

⋅+⎥

⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

⋅=∂∂

ρα

τ&

2

2

2

2

2

2

1-3a

where pc

k⋅

=ρ

α = thermal diffusivity of the material [m2/s]

k = thermal conductivity, ρ =density [kg/m3], cp= heat capacity [J/(kg.°C)] q = internal heat generation per unit volume of material [W/m3] τ = time [s] T = temperature, [K or °C] x, y and z = coordinates [m]

The Fourier equation gives solutions for the temperature fields in most applications in steady state as well as non stationary heat conduction (transient) cases. From the Fourier law follows solutions for a number of important cases. Simple examples are: Heat rate flow by conduction through a wall in steady state:

xTTAkq m

Δ−

⋅⋅=21 2-1

where k = thermal conductivity of the wall material [W/(m2.°C)] T1 – T2 = temperature difference between wall surfaces [K or °C] ∆x = the thickness of the wall Am = wall surface area: (For a flat wall Am is naturally equal to the area)

for a cylindrical wall use: )/ln( 21

21

AAAAAm

−=

for a spherical wall use: 21 AAAm ⋅= A1 and A2 = areas of the outer and inner surfaces of the cylinder or

sphere

Temperature profiles for a cylindrical geometry without internal heat generation:

⎟⎠⎞

⎜⎝⎛⋅

⋅⋅⋅=−

11 ln

2 rr

LkqTTπ

or: )/ln()/ln()(12

1121

rrrrTTTT ⋅−=−

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where q/L = rate of radial heat flow per meter of cylinder length [W/m] T1 and T2 = temperatures on the radius r1 and r2 The temperature profile in a slab (a plane plate or wall) with internal heat generation:

211 )(

2xx

kqTT v

−⋅⋅

−=&

where T and T1 denote temperatures in sections x and x1 x, x1, = distance from the centerline.

qv = rate of internal heat generation per unit volume [W/m3] Heat conduction in a straight rod with heat exchange to the surrounding: The temperature profile is calculated by the following relation:

[ ][ ]Lm

xLm⋅−⋅

⋅=cosh

)(cosh0θθ 2-33b

The tip temperature difference (at x = L) is:2

[ ]Lm ⋅⋅=cosh

10θθ

where θ = temperature difference to surrounding at position x, [°C], θ 0 = temperature difference to surrounding at x = 0, [°C]

L = length of rod [m]

AkPhm

⋅⋅

=

h = heat transfer coefficient at the rod surface [W/(m2.°C)] P = circumference of rod [m] A = cross sectional area of rod [m2] k = thermal heat conductivity of rod material [W/(m.°C)] Fin efficiency:

Z

ZmLf

)tanh(0

==θθη 2-38

where Z = m .L; L = fin length; AkPhm

⋅⋅

=

For straight plane (flat) fins with constant thickness: tk

hm⋅⋅

=2 ; t = fin thickness,

In case of cylindrical pin fin, use: t = d/2; d = diameter of pin fin For straight plane (flat) fins with varying thickness:

In case of flat trapezoidal profile, use: t = ¾ .t”+ 1/4 .t’

t” and t’ thickness of fin at root and top sections In case of conical spine fin, use: t = 9/8 .d” d” diameter at root section (the conical spine ends in a point where d = 0)

For fins on tubes: L = equivalent length of fin, depending on fin shape: ϕ⋅= 0rL r0 = inner radius of fin (half the tube diameter)

ϕ = correction factor:

2 This equation describes for instance the error in temperature measurement caused by conduction in the walls of a temperarture pocket

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For circular fins*): ))ln(35,01()1( ρρϕ ⋅+⋅−≅ ρ = R/r0

R = outer fin radius In case of non constant fin thickness trapezoidal form use t = (t”+t’)/2 where t” and t’ are thickness at root and top section of fin.

For rectangular fins: Relations as for circular fins but with ρ = ρm as follows:

ρm = 1,28.R/r0 .(H/R-0,2)0,5

H/R = aspect ratio (to be chosen so that H>R)

For hexagonal fins: Relations as for circular fins but with ρ = ρm as follows: ρm = 1,27.R/r0

.(H/R-0,3)0,5 H/R = aspect ratio (to be chosen so that H>R)

A diagram summarizing relations and results is found in Appendix “Fin Efficiency” Numerical methods for solving stationary heat conduction problems Several (commercial) computer codes are available. Different simple finite difference methods can however be used. One such method uses a “relaxation” technique as follows: The geometry is divided into a quadratic mesh. Such division means that the temperature in each node, if stationary conditions prevails, in equilibrium must be t0 = (t1+ t2+ t3+ t4)/4 where ti is the temperature in the four adjacent nodes. To solve a problem where the temperature distribution is sought, the calculation starts with guessing temperatures in all nodes. Using the technique of relaxation, the errors of the guessed temperatures are calculated in each node. The guessed temperature field is corrected by successively adjusting each and every node temperature until the errors are negligible small in all node points. The calculations can preferably be executed in for instance an Excel-sheet. To calculate heat conduction in a simple way for given geometries a method of Shape factors can be used. In the Appendix “Shape factors for heat conduction” relations and factors are given for a number of geometries. (Other cases are found in Holman Table 3-1).

Non stationary heat conduction problems without internal heat sources: a) Body with no internal heat conduction resistance (“lumped capacitance method”):

)/exp( 00 ττθθ ⋅= 4-5 where θ = temperature difference at time τ after initiation of the process θ0 = initial temperature difference (at time τ = 0) 0τ = “time constant” = m.cp/(h.A) (m = mass of body; cp= specific heat

capacity; h = heat transfer coefficient; A = surface area.) This method is applicable with good accuracy for Bi = h (V/A)/k < 0,1

b) Body with internal heat conduction resistance:

Simplified case: As shown by Bäckström and Allander (SF-Review 1969) and approximate compensation for internal heat conduction resistance can be made so that the relation 4-5

*) These relations are approximations, based Th E Schmidt, ´Die Wärmeleistung von berippten Oberflächen´, Karlsruhe, 1950.

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can be used with reasonable accuracy. Instead of the external heat transfer coefficient h a corrected value (hcorr) is to be used in expression for the time constant τ 0 = m .cp/(hcorr

.A) as

follows: kz

shhcorr ⋅

+=11

where for: plate cooled on both sides cylinder sphere s = half the slab thickness s = radius s = radius z = 3 z = 4 z = 5

With this approximation equation 4-5 holds with good accuracy for mean temperature differences in the body for cases where the Biot number Bi is < 1. (Bi = h .s/k). General treatment of transient heat conduction problems. Solutions of the Fourier equation, can be given in the following general form:

);;;(0 geometrypositionBiFof⋅= θθ where θ = temperature difference at timeτ after initiation of the process θ 0 = initial temperature difference (at timeτ = 0)

2sFo τα ⋅

= = Fourier number (“dimensionless time”) [-]

kshBi ⋅

= = Biot number. Describes the ratio between internal and external

thermal resistances. [-] α = k/(ρ .cp) = the thermal diffusivity of the material [m2/s] k = thermal conductivity [W/(m K)] h = heat transfer coefficient on body surface [W/(m2.°C)]

s = a characteristic length of the body. (s =Volume/Area of body). For a cylinder or sphere: s = radius, For a plate cooled on both sides: s = half the plate thickness [m]

Holman figures 4-7 to 4-16 give solutions for the surface, center, and average temperatures of a large plate (slab) or a long cylinder. Two diagrams giving equivalent information but arranged in a different way (Pierre, 1982) are shown in Appendix, “Temperatures in transient conduction – Plate” and “Temperatures in transient conduction – Cylinder”. Superposition Solutions can be derived for two- or three-dimensional cases (such as a rectangular bar or short cylinder) by super positioning the solutions for one-dimensional geometries. For instance the center temperature of a short cylinder can be found simply by multiplying solutions for a long cylinder and a large plate (plate thickness equivalent to the length of the short cylinder): plateCcylinderCdershortcylinC )/()/()/( 000 θθθθθθ ⋅= Numerical methods One simple way for calculation transient problems is the Schmidt’s method. It can be used for one-dimensional problems for instance temperature profiles at different time steps in a plate such as a wall. The plate is divided into sections, each with thickness ∆x. In the Schmidt’s method the time step must be chosen to: )2/(2 ατ ⋅Δ=Δ x whereα = the thermal diffusivity of the material 4-48 The temperature in a section ‘m’ at a time ‘p+1’ is estimated from the temperature at the preceding time step ‘p’ in the adjacent sections by the following simple equation:

2/)( ;1;11; pmpmpm ttt −++ +=

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LAWS OF SIMILARITY In order to decrease the number of parameters it is often very effective to utilize laws of similarity, whenever possible. For phenomena where the basic governing laws are known such laws are easy to formulate, for other cases one can sometimes use a so called dimensional analysis.

Heat conduction For heat conduction problems it can (based on the Fourier equation) be shown that the temperature field (θ/θ0) for a given geometry can be characterized by the following dimension-less parameters:

)/;;(/ 0 sxBiFof=θθ

where 2sFo τα ⋅

= = Fourier number; k

shBi ⋅= = Biot number

α = k/(ρ .cp) = the thermal diffusivity of the material [m2/s] k = thermal conductivity [W/(m K)] τ = time [s] s = a characteristic length of the body [m] h = heat transfer coefficient on body surface [W/(m2.°C)] x = position in the body (x/s signifies the position in a dimensionless form). Convective heat transfer without phase change: Velocity and temperature fields in a fluid at a surface for a given geometry (but not necessarily the same size) can be characterized as follows:

velocity field: ux/u0 = f(Re; position x/d; geometry)

temperature field:

θx/θ0 = f(Re; Pr; position x/d; geometry)

where μρ

υdudu ⋅⋅

=⋅

=Re = Reynolds number [-];

kcp⋅

=μPr = Prandtl number [-]

u = velocity [m/s] d = characteristic length [m] ρ = density [kg/m3] μ = dynamic viscosity [Ns/m2] υ = kinematic viscosity [m2/s] ux = velocity at position ’x’ [m/s] u0= velocity in a reference position (or an average velocity) [m/s] θx = Tx - Twall = temperature difference at position ’x’ [°C]

θ0= temperature difference at a reference position ’0’ (or average temperature difference) [°C]

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Pressure drop: From the similarity laws for the velocity fields follows that the pressure difference, ∆p, is:

∆p = ρ .u2 .f(Re; geometry) which also can be expressed by a dimensionless number, the Euler number, Eu:

2upEu

⋅Δ

=ρ

Heat transfer: The similarity of the temperature fields have as a consequence that the heat transfer coefficient, h, can be expressed by a dimensionless number, Nu:

kdhNu ⋅

= = Nusselt number [-]

Where h = heat transfer coefficient [W/(m2.°C)] d = characteristic length, [m] k = thermal conductivity of the fluid [W/(m.°C)] The Nu-number describes the temperature profile in the boundary layer close to the surface. More precisely it is a dimensionless temperature gradient at the surface. For a case with forced convection (heat transfer without change of phase) we have:

Nu = f(Re; Pr; geometry)

For free convection there is no direct way to express the Reynolds number. Instead a Grashof number, Gr, is used by which:

Nu = f(Gr; Pr; geometry)

where 2

3

υβ xTgGr ⋅Δ⋅⋅

= = Grashof number [-]

g = acceleration of gravity, (normally g = 9,81 m/s2) β = temperature coefficient of volume expansion. For an ideal gas β = 1/T where T = gas absolute temperature [K] ∆T = Tw – T = temperature difference [K or °C] x = characteristic length [m] υ = kinematic viscosity of the fluid [m2/s]

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FORCED CONVECTION WITHOUT CHANGE OF PHASE. PRESSURE DROP AND HEAT TRANSFER .

PRESSURE DROP FOR FLOW IN CHANNELS Due to friction a pressure drop will occur when a media flows in a channel. The pressure drop, Δp, in a channel of length L with a hydraulic diameter dh can be estimated by:

hdLufp ⋅

⋅⋅=Δ

2

2ρ

where f = friction factor, ρ is fluid density and u is the fluid flow velocity dh = hydraulic (or equivalent) diameter.

For non-circular channels it is proposed to use an “equivalent diameter”, dh,

defined as: P

Adh⋅

=4

where A = cross sectional area of channel P = circumference of channel. For turbulent flow the friction factor as given on page 15 can be used.

For fully developed laminar flow (for long tubes, x > 0,05 .dh.Re) the local friction factor is

Re/Cf =

where μρ

υhh dudu ⋅⋅

=⋅

=Re = the Reynolds number

for circular tubes: C = 64 square ducts: C = 56,9 triangular ducts: C = 53,3 flat ducts C = 96 The friction factor is somewhat higher in the entrance region (for x < 0,05 .dh

.Re).

HEAT TRANSFER FOR FLOW IN CHANNELS

Turbulent flow Reynolds’ analogy is based on similarities between heat transfer and fluid friction (which causes pressure drop). The simple analogy is valid only for fluids with Pr = 1. For such cases it is shown that in non-dimensional form:

8/PrRe

fNu=

⋅ 6-8

Where Nu = kdh ⋅ = the Nusselt number;

kcp⋅

=μPr = the Prandtl number

f = friction factor

The Reynolds´ analogy (equation 6-8) can also be expressed directly as:

u

pcAAh p

Q

w Δ⋅⋅= [W/(m2.°C)]

u = flow velocity [m/s] ∆p = pressure drop [Pa] in a channel with length L [m] and diameter d [m] cp = specific heat capacity of the fluid [J/(kg.°C)] Aw = sectional flow area of channel (if circular = 4/2d⋅π ) [m2] AQ = heat transfer area of channel (if circular = Ld ⋅⋅π ) [m2]

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Notice that the analogy by Reynolds only holds for fluids with Pr = 1. For other cases a Pr-number correction is to be included. The Prandtl equation is a development of the Reynolds´ analogy. In Prandtl’s equation the phenomena in the boundary layer close to the surface are also considered and this makes the result valid also for fluids with Pr different from 1:

)1(Pr1

8/PrRe −⋅+

=⋅ ϕ

fNu

where ϕ = ub/um = velocity ratio

With traditional relations for the friction factor, f, as given by Blasius (f = 0,316/Re0,25) the result is the ´Prandtl-Hofmann´s equation´:

)1(Pr1PrRe079,0 75,0

−⋅+⋅⋅

=ϕ

Nu

According to Hofmann the velocity ratio ϕ, for fluids 0,74<Pr<300, can be expressed by: )6/1()8/1( PrRe5,1 −− ⋅⋅=ϕ Using instead a relation recommended by Mc Adams for the friction factor:

2,0Re184,0

=f

and a simplified correction for the influence of the Pr-number the result is the following widely used relation (sometimes called the Dittus Boelter equation):

4,08,0 PrRe023,0 ⋅⋅=Nu

The relation is valid for relatively long channels with L/d > 60. Fluid properties are to be determined at the bulk temperature. It is valid for fluids with Pr > 0,7 and for Re > 10000 (for fluids with relatively low viscosity (μ < μwater) it can often be extrapolated to about Re> 3000).

For shorter channels (L < 60 .d) a correction is often recommended by a factor =05,0

/60

⎟⎠⎞

⎜⎝⎛

dL.

Other closely related relations are also often used, such as: 2,03/2 Re023,0Pr −⋅=⋅St

where pcu

hNuSt⋅⋅

=⋅

=ρPrRe

= the Stanton’s number

fluid properties are to be determined at ´film´ temperature, Tf = (Tw+Tb)/2

Still another similar relation to take varying viscosity of the fluid into account is: 2,014,03/2 Re023,0)/(Pr −⋅=⋅⋅ bwSt μμ where μw and μb = dynamic viscosity of fluid at wall and bulk temperatures all other properties at bulk temperature. A widely used relation proposed by Gnielinsky (Int. Chem. Eng.Vol 16 p 359. 1976) which correlates data in a wide range (0,5 < Pr < 2000 and 2300 < Re < 5 . 106) is the following:

)1(Pr)8/(7,121

Pr)1000(Re)8/()3/2(5,0 −⋅⋅+

⋅−⋅=

ffNu

where for smooth tubes: { }264,1ln(Re)79,0

1−⋅

=f

16

The relations are normally valid also for non-circular channels where an “equivalent

diameter”, dh, is used. As previously defined: P

Adh⋅

=4

where A = cross sectional area of channel and P = circumference of channel. For “concentric tubes”, where the fluid flows in a ring-shaped channel with inner diameter d1 and outer diameter d2 the following relation is given by Wiegand (valid for 1<d2/d1<10):

)3/1(8,0 PrRe023,0 ⋅⋅⋅= ϕNu

Where ϕ = 1 if the heat transfer takes place at the outer surface (at d = d2) ϕ = (d2/d1)0,45 if the heat transfer takes place at the inner tube (at d = d1)

As equivalent diameter in Nu and Re use: dh = 4.A/U = d2 - d1

Laminar flow A classical solution for fully developed laminar flow in circular channels was presented by Graetz. The solution is based on the assumption that the velocity profile is fully developed already where the heated (or cooled) section starts and from that point on, the tube surface is assumed to be at a constant temperature. The temperature profile in the fluid is given by the Graetz solution in the following form:

)/;(/ ww rrGzf=θθ Where θ and θw = temperature differences on radius r and rw xdGz /PrRe ⋅⋅= = Graetz number3 6-15 x = distance from the start of the heated section.

The Graetz solution shows that the temperature profile becomes fully developed after an entrance region. The entrance region covers a distance, xt:

xt = 0,05.Re.Pr.d. A consequence of the Graetz solution is that, for x>xt, the local Nu-number is constant (only a function of the shape of the channel). For circular channels with constant temperature it is found that

Nux>xt = 3,66 For other geometries and other boundary conditions the local Nu-number will be different. The following table give results for two different boundary conditions, constant temperature (“T”) and constant heat flux (”q”): “T” “q” Circular tube: Nux>xt = 3,66 4,36 Equilateral triangle: 2,47 3,11 Quadratic channel: 2,98 3,61 Rectangular channel, aspect ratio = 8: 5,60 6,49 Duct formed by two wide parallel walls: 7,54 8,24 The following equation attributed to Hausen is applicable to long as well as short circular channels with constant wall temperature:

[ ] )3/2(PrRe)/(04,01

PrRe)/(0668,066,3⋅⋅⋅+

⋅⋅⋅+=

LdLdNu 6-9

3 There are different definitions of the Graetz number. Some references use PrRe

4⋅⋅

⋅⋅

=⋅⋅

=xd

xkcmGz p π&

17

For flow in channels with (Re.Pr.d/L) > 10 results from the theory can be summarized by the following simple equation: )3/1()/Pr(Re62,1 LdNu ⋅⋅⋅= Experimental results are reported about 15% higher and in an equation by Sieder and Tate (Ind. Eng. Chem. Vol. 28, 1429, 1936) the following correlation is recommended where also the influence of temperature dependent viscosities are taken into account: 14,0)3/1( )/()/Pr(Re86,1 wLdNu μμ⋅⋅⋅⋅= where μ and μw denote the fluid dynamic viscosity at bulk and wall temperatures. In this case, the heat transfer coefficient is based on the arithmetic mean of the inlet and outlet temperature differences. FORCED FLOW OVER EXTERIOR SURFACES.

Friction and Heat Transfer for fluid flow over plane surfaces.

To characterize the flow over a surface a Reynolds number is used defined as:

υ

xux

⋅=

∞Re

where u∞ = undisturbed velocity [m/s] x = characteristic length. For flow along a plane surface = distance from starting

edge of the surface A boundary layer will be formed close to the surface, starting at x = 0. The boundary layer will initially be laminar but changes from laminar to turbulent character after a distance x = xtrans. The transition occurs generally at approximately Recrit = 5.105

(occurs hence at a distance x = xtrans = 5.105.υ/u∞ from the starting edge). For cases where the velocity and temperature fields both start at x = 0 we have: Within a laminar boundary layer (Rex < 5.105): Boundary layer thickness (99%): 5,0)(Re64,4 −⋅⋅= xs xδ 5-21 Local shear stress, fluid to surface: 5,02 )(Re)(323,0 −

∞ ⋅⋅⋅= xw uρτ

Local heat transfer coefficient, hx, defined by local Nu-number,k

xhNu xx

⋅= :

)3/1(5,0 Pr)(Re332,0 ⋅⋅= xxNu 5-444 The average heat transfer coefficient between x = 0 and x = L is twice the local coefficient at x = L, giving: )3/1(5,0 Pr)(Re664,0 ⋅⋅= LLNu 5-46b

With fully turbulent boundary layer (Rex > 5.105): Boundary layer thickness (99%): 2,0)(Re381,0 −⋅⋅= xs xδ 5-91 Local shear stress, fluid to surface: 2,02 )(Re)(0288,0 −

∞ ⋅⋅⋅= xw uρτ

Local Nu-number: k

xhNu xx

⋅= )3/1(8,0 Pr)(Re0296,0 ⋅⋅= xxNu 5-81

The average heat transfer coefficient between x = Lcrit and x = L is: )3/1(8,0 Pr)(Re036,0 ⋅⋅= LLNu The average heat transfer coefficient between x = 0 and x = L, for the laminar as well as the turbulent regions integrated for a flat plate (provided that transition occurs at Retrans = 5.105) is approximately: [ ] )3/1(8,0 Pr835)(Re036,0 ⋅−⋅= LLNu

4 The difference of about 3% in the numerical constant in the expressions for shear stress and Nu-number (0,323 and 0,332) is the result of the approximate nature of the integral boundary layer analysis.

18

Flow across single cylinders and tube bundles. The heat transfer on the surface of a single tube with fluid flow perpendicular to its axis can be estimated by: 33,0PrRe ⋅⋅= nCNu 6-17

where kdhNu /⋅= d = the tube diameter [m] υ/Re du ⋅= ∞ u∞ = undisturbed fluid flow [m/s]

Properties to be taken at film temperature (= (tw+ t∞)/2)

C and n can be taken from the following table: 1 < Re < 4 C = 0,989 n = 0,33 4 < Re < 40 0,911 0,385 40 < Re < 4000 0,683 0,466 4000 < Re < 4.104 0,193 0,618 4.104 < Re < 4.105 0,0266 0,805

In tube bundles for 2000 < Re < 40000 the Nu-number can be estimated by eq. 6-17 where υ/Re max du ⋅= umax = velocity in the most narrow section between tubes in the tube bundle C and n are functions of the bundle geometry, ST/d and SL/d. See Table. ST = the distances between tube centers perpendicular to the flow [m]

SL = the distances between tube centers along the flow [m] d = tube diameter [m]

TABLE: Constants C and n for tube bundles (equation 6-17) (Holman table 6-4) SL/d= ST/d = 1,25 ST/d = 1,5 ST/d = 2 ST/d = 3 Tubes in line C n C n C n C n 1,25 0,386 0,592 0,305 0,608 0,111 0,704 0,0703 0,752 1,5 0,407 0,586 0,278 0,620 0,112 0,702 0,0753 0,744 2,0 0,464 0,570 0,332 0,602 0,254 0,632 0,220 0,648 3,0 0,322 0,601 0,396 0,584 0,415 0,581 0,317 0,608 Staggered tubes 0,6 0,236 0,636 0,9 0,495 0,571 0,445 0,581 1,0 0,552 0,558 1,125 0,531 0,565 0,575 0,560 1,25 0,575 0,556 0,561 0,554 0,576 0,556 0,579 0,562 1,5 0,501 0,568 0,511 0,562 0,502 0,568 0,542 0,568 2,0 0,448 0,572 0,462 0,568 0,535 0,556 0,498 0,570 3,0 0,344 0,592 0,395 0,580 0,488 0,562 0,467 0,574

Flow over spheres Flow over spherical bodies of air with 17 < Re < 70000 give: 6,0Re37,0 ⋅=Nu 6-25 where kdhNu /⋅= d = the tube diameter [m] υ/Re du ⋅= ∞ u∞ = undisturbed fluid flow [m/s] Another relation with a wider range of validity, 1 < Re < 200000, is: 54,025,03,0 Re53,02,1)/(Pr ⋅+=⋅⋅ − μμwNu 6-29 where properties are at undisturbed fluid temperature, except μw at wall temperature.

19

HEAT TRANSFER IN FREE CONVECTION Relations for heat transfer in free convection are in non-dimensional form:

Nu = f(Gr; Pr, geometry)

where kHhNu /⋅= Pr = μ .cp/k

2

3

υβ HTgGr ⋅Δ⋅⋅

= = Grashofs number

β = temperature coefficient of volume expansion for the fluid, defined as

Tv

v ∂∂

⋅=1β where v = specific volume [m3/kg]. For ideal gas

T1

=β [1/K]

g = acceleration due to gravity [m/s2] ∆T = temperature difference between wall and fluid [K or °C] H = wall height [m]

If viscous forces are dominating and inertia forces are ignored it can be shown that: Nu = f(Gr . Pr) If instead the viscous forces are small compared to inertia forces the result is: Nu = f(Gr . Pr2)

Free convection on vertical plates The theoretical results, in good agreement with experiments, are summarized by:

Case 0) 1 < Gr . Pr < 104 Nu = 1,35 . (Gr . Pr)0,15 Case 1) 104 < Gr . Pr < 108 Nu = 0,56 . (Gr . Pr)(1/4) 7-25 Case 2) 108 < Gr . Pr < 1012 Nu = 0,13 . (Gr . Pr)(1/3) Fluid properties are to be determined at “film temperature” = (Twall+Tamb)/2 Comment:

For most cases the flow on a vertical surface will have a laminar region at the start of the convection with a transition to turbulent boundary layer, often around Gr . Pr = 108. For Gr . Pr > 108 there is thus a region with laminar as well as one with turbulent flow in the boundary. Considering the two cases 1) and 2) the average value of the Nu-number on the whole plate can be estimated by: Nu = 56 + 0,13 . {Gr . Pr(1/3) – 464} applicable for 108 < Gr . Pr < 1012

The equation given for ‘Case 1)’ corresponds to a laminar flow in the boundary layer. This relation can be simplified to:

)4/1(

⎥⎦⎤

⎢⎣⎡Δ

⋅=HTKh L

The ‘Case 2)’ corresponds to turbulent flow in the boundary layer and this relation can in a similar fashion be simplified to: )3/1(TKh T Δ⋅= Values of the constants KL and KT for air and water are given in the Table on the following page.

20

Free convection on vertical cylinders For large diameters, d, in relation to the cylinder height, H, the relations for vertical plates are used. In cases with small ratios d/H the heat transfer is enhanced compared to that for the vertical plate. For a case with H = d, and for 104 < Gr . Pr < 106, the heat transfer can be estimated by Nu = 0,775 . (Gr . Pr)0,21 (Holman Table 7-1) Free convection on horizontal cylinders The following relations are recommended by Pierre:

104 < Gr . Pr < 108 Nu = 0,50 . (Gr . Pr)(1/4) 108 < Gr . Pr Nu = 0,11 . (Gr . Pr)(1/3)

The characteristic length in Nu and Gr is the diameter d, of the cylinder. Fluid temperatures are to be taken at “film temperature” = (Twall+Tamb)/2.

The equations for horizontal cylinders can be written in similar simplified form as for vertical plates. Applying the data for KL and KT as given in the following table for vertical plates the results for horizontal cylinders for laminar and turbulent cases are:

)4/1(

89,0 ⎥⎦⎤

⎢⎣⎡Δ

⋅⋅=dTKh L and )3/1(85,0 TKh T Δ⋅⋅=

TABLE

tF 3

PrHT

Gr⋅Δ⋅ KL KT

°C 1/(K m3) SI-units to be used Air: -150 615 . 107 1,89 2,85 For vertical plates: -100 115 . 107 1,70 2,25 If 104 < Gr . Pr < 108 then: -50 34,8 . 107 1,57 1,88 h = KL . (∆T/H)(1/4) 0 14,5 . 107 1,49 1,66 50 6,75 . 107 1,41 1,48 If 108 < Gr . Pr < 1012 then: 100 3,47 . 107 1,35 1,33 h = KT . ∆T (1/3) 200 1,18 . 107 1,27 1,14 300 5,1 . 106 1,21 1,01 400 2,54 . 106 1,15 0,91 600 0,85 . 106 1,06 0,76 Water: 10 0,48 . 1010 85 127 20 1,40 . 1010 115 187 40 3,79 . 1010 155 274 60 6,82 . 1010 188 346 80 10,5 . 1010 213 408 100 14,9 . 1010 237 468 140 25,8 . 1010 272 566 180 41,3 . 1010 302 655 220 62,4 . 1010 325 727

21

Free convection on horizontal plates The following relations can be used for cases of a heated plate facing upward or a cold plate facing downward: 105 < Gr . Pr < 108 Nu = 0,54 . (Gr . Pr)(1/4) 108 < Gr . Pr Nu = 0,14 . (Gr . Pr)(1/3)

The characteristic length in Nu and Gr is the side length of the plate. The relations have been determined essentially for quadratic plates. For rectangular plates the average length of the two sides may be used.

In case of a heated plate facing downward or a cold plate facing upward the simple rule of thumb is that the heat transfer coefficient is about half that for the opposite case. For such cases Mc Adams give the following relation: 3 . 105 < Gr . Pr < 3 . 1010 Nu = 0,27 . (Gr . Pr)(1/4)

A special problem arises for water around +4°C since water has a density maximum at this temperature.

Free convection between surfaces in enclosed spaces It is convenient to define a heat transfer coefficient h’ = q/(T1 - T2) based on the temperature difference between the two adjacent surfaces, (T1 - T2) in the enclosed space. (Notice that h’ is in effect combining heat transfer resistances at both surfaces. It is in thus similar to a ´U´-value for the air space.) The following definitions are used:

khNu δ⋅

='' and 2

3

υδβ ⋅Δ⋅⋅

=TgGr

where δ is the distance between two walls of the enclosure and ∆T = T1 - T2 is the temperature difference between the walls. For air filled enclosures the following relations may be used

Horizontal air spaces (heat flow in the upward direction):

104 < Gr < 4 . 105 Nu’ = 0,195 . Gr (1/4) 4 . 105 < Gr Nu’ = 0,068 . Gr (1/3)

Vertical air spaces: (typically between two glasses in a double glazed window)

2 . 104 < Gr < 2 . 105 Nu’ = 0,18 . Gr (1/4) . (H/δ )-(1/9) 2 . 105 < Gr < 1,1 . 107 Nu’ = 0,065 . Gr (1/3) . (H/δ )-(1/9)

H = height of the enclosure δ = distance between the walls of the enclosure.

22

Free convection heat transfer in vertical open channels In channels open at top and bottom a flow of fluid will be induced by free convection if there is a temperature difference. The following relations are given by Elenbaas:

)4/3(

)/(35exp1

241* ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−−⋅⋅⋅=LsRaL

sRaNus

s

where Nu* = h* . s/k

Pr)(2

3

⋅⋅−⋅⋅

=∞

υβ sTTgRa s

s

h* = heat transfer coefficient based on temperature difference (Ts - T∞) where Ts is the surface temperature and T∞ is the inlet gas (fluid) temperature

s = distance between walls L = length (height) of channel

HEAT TRANSFER IN FALLING FILMS For falling films on a vertical plate it is shown that the Reynolds number is determined by:

μ

Γ⋅=

&4Re

where Γ& = mass flow rate divided by the breadth of the plate [kg/(m.s)] μ = dynamic viscosity of the liquid in the falling film

For Re < 1600 the film will usually be laminar.

The thickness,δ , of a falling film in laminar flow on a vertical surface (in equilibrium) is:

)3/1(

3⎥⎦

⎤⎢⎣

⎡⋅

Γ⋅⋅=

ρυδ

g

&

where υ and ρ = the kinematic viscosity and the density of the fluid g = acceleration due to gravity (normally g =9,81 m/s2) The temperature in the film along the plate can be described by a factor ϕ which represents the heat exchanger effectiveness:

0/θϕ bTΔ=

where ∆Tb = bulk temperature change from the inlet (=Tin - Tb) θ0 = temperature difference at inlet section, at top of the wall (=Tin - Twall)

A solution for the temperature field as given by Nusselt is approximately:

For S < 0,05: 656,023,2 S⋅=ϕ S > 0,05: )65,5exp(9101,01 S⋅−⋅−=ϕ

where 4δρυ

⋅⋅⋅⋅⋅

=gc

xkSp

x = distance from inlet

23

The result of the Nusselt solution can also be expressed in form of heat transfer coefficients and represented by the following dimensionless parameters (as introduced by Pierre, Kylteknisk Tidskrift Vol 21, p 101 – 110, Dec 1962):

k

HhNu aa

⋅=

kcp⋅

=μPr = Prandtl number of fluid in film

2

3

υxgG ⋅

= 4

PrRe( ⋅=

⋅Γ=

kcGz p&

= modified Graetz number)

According to the treatment the heat transfer coefficient (ha based on the arithmetic average temperature difference between temperatures at inlet and at distance x from inlet) expressed by means of the Nua-number becomes:

For 1/S < 1,5 Nua = 2 . Gz (which is equivalent to ϕ = 1) 3< 1/S < 102 )4/1(Pr)(19,1 ⋅⋅= GNua

102 < 1/S < 104 )9/2()9/1( Pr)(915,0 ⋅⋅⋅= GGzNua

It is to be noticed that the middle equation (for 3< 1/S < 102) is valid for cases where the “temperature efficiency” ϕ is between 0,07 and 0,88, which covers most cases of interest in applications with laminar films. The results are also represented in a diagram given in the Appendix Heat Transfer in Falling Films / Laminar film.

For a not perfectly vertical plate with an inclination by an angle φ from the horizontal plane the equations apply as given but g in the equations is to be replaced by ( )sin(φ⋅g ) In turbulent flow, for Re>1600, the film will become thicker than according to the equation for laminar flow. (This is due to increased apparent viscosity due to the turbulence.) The following equation may be used for the film thickness for Re>1600:

)5/1()3/1(

1600Re3

⎥⎦⎤

⎢⎣⎡⋅⎥

⎦

⎤⎢⎣

⎡⋅

Γ⋅⋅=

ρυδ

gt

&

The heat transfer coefficient in turbulent film flow is (according to results from tests with water, here in generalized form) given by: )3/1()(0159,0 GzGNu ⋅⋅= The relations for turbulent films are shown in the diagram Appendix Heat Transfer in Falling Films / Turbulent film. The coefficient of heat transfer in falling films on horizontal cylinders (tubes) is given by: 1/Sh < 4 Nua = 2 . Gz 7,5 < 1/Sh < 2.102 )4/1(Pr)(02,1 ⋅⋅= GNua

2.102 < 1/S < 104 )9/2()9/1( Pr)(82,0 ⋅⋅⋅= GGzNua

where 4hp

hgc

HkSδρ

υ⋅⋅⋅

⋅⋅= (or )3/4(3/1 )22,7(Pr)( −⋅⋅⋅= GzGSh )

kc Pr HgG p

2

3 ⋅=

⋅=

⋅Γ=

⋅=

μυk

cGzk

HhNu paa

&

H = π.d/2 = half the circumference of the cylinder Γ& = mass flow rate per cylinder half and per meter of cylinder length.

24

HEAT TRANSFER IN CONDENSATION When a vapor is exposed to a surface of a lower temperature than the temperature corresponding to the saturation temperature, the vapor will condense into liquid form on the surface. (Notice the saturation temperature is determined by the pressure of the vapor; vapor temperatures higher than saturation have virtually no influence.) Generally the liquid formed by condensation collects into a film on the condensation surface; so called film condensation.

The classical theory by Nusselt postulates that there is no thermal resistance for the condensation process itself. The sole thermal resistance for film condensation is the resistance introduced by the liquid film. Thus, provided that the film is laminar, the heat transfer coefficient is simply based on the thermal conduction through the film:

δkh =

where k = the thermal conductivity of the condensed liquid δ = thickness of the film. Condensation on vertical surfaces

In laminar film condensation on a vertical wall, the film thickness δ will increase by the distance from the top of the wall, x, due to the accumulation of condensate:

)4/1(

4⎥⎦

⎤⎢⎣

⎡⋅⋅

⋅Δ⋅⋅⋅=

ρυδ

ghxTk

fgx 9-6

where ∆T = temperature difference between the saturation temperature of the vapor and the wall temperature [°C]

υ = kinematic viscosity of the condensed liquid [m2/s] hfg = latent heat of condensation of the fluid [J/kg] ρ = density of the condensed liquid (the density of the vapor is here assumed

to be << the density of the liquid, (otherwise replace ρ by: (ρliquid -ρvapor)) [kg/m3]

g = acceleration due to gravity (normally g = 9,81 m/s2)

Inserting this expression for the film thickness into the equation for heat transfer gives the local heat transfer coefficient, hx:

)4/1(3

4 ⎥⎦

⎤⎢⎣

⎡⋅Δ⋅⋅⋅⋅⋅

=xT

ghkh fgx

υρ 9-7

The average value (h) of the heat transfer coefficient on a wall with condensation from x = 0 to x = length L, is found by integration of the local values of hx from x = 0 to x = L. The result is simply that h = 4/3 . hx=L:

or: )4/1(3)4/1(3

943,043

4⎥⎦

⎤⎢⎣

⎡⋅Δ⋅

⋅⋅⋅⋅=⎥

⎦

⎤⎢⎣

⎡⋅Δ⋅⋅⋅⋅⋅

⋅=LT

ghkLT

ghkh fgfg

υρ

υρ 9-10

where L = condensing length on the vertical plate. The average heat transfer coefficient can also be expressed in dimensionless form by: )4/1(943,0 CvNu ⋅=

where TkLgh C

kLhNu v

Δ⋅⋅⋅⋅⋅

=⋅

=υ

ρ 3fg and (a dimensionless number)

25

In many cases the temperature difference in condensation (∆T) is not known, but rather the heat flux, q/A [W/m2]. Since q/A is equal to (h . ∆T) the following relation can be deduced:

)3/1()3/1(3

)3/4(

)/(924,0

)/(4)

34( ⎥

⎦

⎤⎢⎣

⎡⋅⋅

⋅⋅⋅⋅=⎥

⎦

⎤⎢⎣

⎡⋅⋅⋅

⋅⋅⋅⋅=

LAqghk

LAqghkh fgfg

υρ

υρ

The relations given apply for laminar flow in the condensate film. This is the case if approximately Re<1800. As deduced for falling films the Reynolds number is given by:

μΓ&⋅

=4Re

where μ = dynamic viscosity of the liquid [m2/s] =Γ& mass flow rate in the condensate film per breath of the wall [kg/(m s)] (at x = L of the wall the condensate mass flow rate is fghLAqΓ /)/( ⋅=& )

For condensation on an inclined wall the equation applies as given if the acceleration due to gravity (g) is replaced by g . sin(φ ) where φ is the angle of inclination to a horizontal plane (if the wall is vertical then φ = 90°). The superheat of the vapor will increase the coefficient of heat transfer somewhat. The influence is simply estimated by replacing the latent heat of condensation of the vapor (hfg) in the equations given and instead use (hfg +cpv

.∆Ts) where cpv is the specific heat capacity of the vapor and ∆Ts = the vapor superheat. The velocity of vapor, uv, will affect film thickness and thereby the heat transfer. If vapor flow velocity is downwards it will increase the heat transfer coefficient, while it is unfavorable if directed upwards. For cases where the vapor flows downwards the following approximate relations can be used:

for X > 1 to 2: )3/1()3/1(

423

⎥⎦⎤

⎢⎣⎡⋅⎥⎦

⎤⎢⎣⎡⋅=

CvLaNu

where TkLgh C

kLhNu v

Δ⋅⋅⋅⋅⋅

=⋅

=υ

ρ 3fg and as earlier

2

31

vv u

gfa ⋅⋅⋅

=ρρ f1 = f/2 = friction factor between vapor and film

ρv = is the vapor density and ρ = density of liquid phase.

)4/1()4

(CvLaX ⋅=

Experimental results on condensation are generally in good agreement with the theory of Nusselt, but often experiments results give somewhat higher heat transfer coefficients which can be attributed to disturbances in the falling film of condensate. Some references recommend that the constant in equation 9-10 may be increased by up to 20%.

With turbulent condensate film (for Re>1800) experimental results give:

Nu = 0,0030 . G . Cv(-1/2)

where 4Re 3

2

3

TkLghCvLgG

kLhNu fg

Δ⋅⋅⋅⋅⋅

=⋅

=Γ⋅

=⋅

=υ

ρυμ

&

26

Condensation on tubes The condensate film formed on a horizontal cylinder will be exhibited to different angles of inclination from top to bottom. Integration along the film in laminar flow will give:

)4/1(3

725,0 ⎥⎦

⎤⎢⎣

⎡⋅Δ⋅

⋅⋅⋅⋅=

dTghkh fg

υρ

where d = the diameter of the horizontal tube and all other symbols are defined earlier. As earlier mentioned it is sometimes more convenient to use the heat flux q/A = h

.∆T instead of the temperature difference. The equation then takes the form:

)3/1(

)/(651,0 ⎥

⎦

⎤⎢⎣

⎡⋅⋅

⋅⋅⋅⋅=

dAqghkh fg

υρ

Condensation inside tubes

Different flow patterns occur in different flow regimes. If the vapor velocity in the tube is low the flow pattern is “stratified”. For condensation inside horizontal tubes the condensate will collect at the bottom of the tube to be drained at the tube exit end. The quantity of condensate at the bottom of the tube can be characterized by a film angle φ m. The condensation will take place with film starting at the top of the tube (φ = 0) and the condensate follows the inside wall until it reaches the bottom condensate at φ =φ m. For this process the heat transfer can be estimated in a similar way as for condensation on the outside of a horizontal cylinder. The portion of the tube inside that is occupied of the condensate at the bottom must however be regarded as insulated and the contribution to the heat transfer at this region neglected since the thermal resistance in the liquid layer is large. Hence the heat transfer can be estimated by integrating the solution for local heat transfer coefficient on horizontal cylinders from φ = 0 to φ =φ m. The results can be expressed by introducing a correction factor F(φ m) as follows (the heat transfer coefficient is based on the full tube inside area):

)4/1(3

725,0)( ⎥⎦

⎤⎢⎣

⎡⋅Δ⋅

⋅⋅⋅⋅⋅=

dTghkFh fg

minsidecylυ

ρφ

The factor F(φ m) is = 1 at φ m = 180° (equivalent to an empty tube, without condensate at the bottom) and F(φ m) = 0 at φ m = 0° (equivalent to an tube filled with condensate), Approximately the relation for F(φ m) can be estimated by

F(φ m) = 1,263 . (φ m/180) – 0,263 . (φ m/180)3

where φ m= is the film angle expressed in degrees (0 < φ m <180). For many cases where the tube is well drained at the outlet a normal value of the angle φ m in practice is about 120° which gives F(φ m) = 0,76. It should be noticed that the arrangement to drain the liquid at the exit of the tube can have a large influence. “Drop wise condensation” occurs if the condensed liquid does not wet the heat transfer surface. The phenomenon is then that no film is formed. Instead liquid droplets forms on the heat transfer surface and they roll more easily off, resulting in only a small thermal resistance. Extremely high heat transfer coefficients can be attained in drop wise condensation due to this. The problem to take advantage of this in engineering applications, is to ensure that this phenomenon will remain for long time in practical situations.

27

Recent models for condensation in tubes have been developed by Cavallini et al 5.

For condensation in horizontal smooth tubes with diameters > 3 mm the following relations are given: For JG>JG

T Heat transfer in annular flow (independent of temperature difference): [ ]1,0144,22363,03685,0817,0 Pr)/1()/()/(128,11 −⋅−⋅⋅⋅⋅+⋅= LLGGLGLLOan xhh μμμμρρ For JG ≤ JG

T Heat transfer in stratified-wavy flow: [ ] strat

TGGstratG

TGAD hJJhJJhh +⋅−⋅= )/()/( 8,0

where[ ]{ } [ ] LOLfgGLLLstrat hxtdhgkxxh ⋅−+Δ⋅⋅⋅⋅−⋅⋅⋅−⋅+⋅=

−)1()/()(/)1(741,01725,0 087,025,0313321,0 μρρρ

dkNuh LLOLO /⋅= where ⋅⋅⋅= 4,08,0 PrRe023,0 LLOLONu ( )[ ] 5,0/ GLGG dgGxJ ρρρ −⋅⋅⋅⋅=

[ ]{ } )3/1(33111,1 )13,4/(5,7

−−−

++⋅= TttT

G CXJ CT = 1,6 for hydrocarbons CT = 2,6 for other refrigerants [ ] 9,05,01,0 /)1()/()/( xxX LGGLtt −⋅⋅= ρρμμ

ReLO=G .d/ Lμ , subscript L refers to liquid; LO to liquid with total mass flow rate PrL = Prandtl number of liquid G= mass velocity [kg/(m2 s)] x = vapor mass quality g = acceleration due to gravitation [=9,81 m/s2]

Lρ and Gρ density of liquid and gas [kg/m3] μ L and μ G dynamic viscosity of liquid and gas [N.s/m2] kL = thermal conductivity of liquid [W/(m °C)] hfg = latent heat [J/kg] Δ t = (tsat - twall) = temperature difference d = diameter of tube ( or hydraulic diameter) [m]

The following relation is given by Cavallini et. al. for condensation of refrigerants in minichannels6 (diameter 1,4 mm) where the shear between vapor and condensing film is dominating and the liquid film is in annular flow (han = Nuan

.d/kL): [ ] 08,01,0

L144,22363,03685,0817,0 )1/(Pr)/1()/()/(128,11 ExNuNu LGGLGLLOan −⋅−⋅⋅⋅⋅+⋅= −μμμμρρ

where: 4,08,0 PrRe023,0 LLOLONu ⋅⋅= (subscript LO refers to liquid with total mass flow rate) [ ]42 10)/()/(log44,0015,0 ⋅⋅⋅⋅+= σμρρ GLLgc jE = entrainment ratio

[ ]xExGgc /)1(1 ⋅−+⋅= ρρ σ = surface tension [N/m] jG = G·x/ρG (superficial gas velocity) Other symbols same as above.

5 Cavallini et al,. Condensation of refrigerants in smooth tubes: A new heat transfer model for heat exchanger design, Proc. HEFAT2004, 3rd Int. Conf. on Heat Transfer, Fluid Mechanics and Thermodynamics. 21-24 June 2004, Cape Town, South Africa. 6 Cavallini et al,. A model for condensation inside minichannels, Proc. HT 05, National Heat transfer Conference, July 17-22, 2005, San Francisco, Calif. USA

28

HEAT TRANSFER IN BOILING LIQUIDS Boiling usually is associated with the formation of vapor bubbles on a heated surface covered by the liquid. The formation of bubbles involves very complex phenomena. A simple but important example considering a bubble (diameter = d) is the following: Due to the surface tension, σ, at the vapor and liquid interface the pressure in a bubble will be somewhat larger than the pressure in the surrounding liquid. A simple force balance

applied on a bubble gives: 4

)(2dppd vb

⋅⋅−=⋅⋅πσπ by which:

dpp vb

σ⋅=−

4

This means that the pressure in the bubble is higher than in the surrounding liquid. A certain superheat, ∆t, is thus necessary to keep a bubble ”inflated”. By using the Clapeyron equation the required superheat to compensate for the pressure difference is found to be:

"

4ρ

σ⋅⋅⋅⋅

=Δfghd

Tt

where T = absolute temperature [K] hfg = latent heat of vaporization [J/kg] ρ” = vapor density [kg/m3] d = bubble diameter [m]

The required superheat is inversely proportional to the bubble diameter, d. For very small diameters, quite large temperature differences (or superheat ∆t) are necessary, often giving an extra resistance in nucleate boiling. Bubbles forms likely at nucleus on the surface where the bubble diameter d is not infinitely small.

BOILING REGIMES When the heated surface is submerged below the free surface of a liquid the process is referred to as pool boiling. Different boiling regimes can be distinguished: pure convection, nucleate boiling and film boiling. The most important region is nucleate boiling where the process is associated with the formation of vapor bubbles on the surface. The pure convective region is limited to small heat flux; the film boiling on the contrary to large heat flux where the bubbles form a gas film on the heated surface.

The Nukiyamas boiling curve (Holman Fig. 9-3) illustrate the different regimes and indicate that there is a “burnout” point. For heat flux above this point a more or less stable vapor film will occur on the hot surface (similar to the Leidenfrost phenomenon). The heat transfer coefficient drops drastically and for a case with given heat flux a very large temperature difference is required in order to transfer the heat from the surface. At these conditions heat is transferred from the surface by radiation and conduction through the vapor film. The large temperature difference required (if the heat flux is fixed) means that the surface temperature often is high enough to melt the surface material (which the term ”burnout” indicates).

The Nukiyamas boiling curve starts at low heat flux, in a convective region. Here the evaporation takes place at the liquid surface, without any formation of bubbles. This applies only for small heat flux to the surface. The evaporation process itself (at the surface) does not require any temperature difference and the liquid surface has a temperature equivalent to the saturation temperature (in similar fashion as in condensation). The thermal resistance lies in the liquid layer between the heat transfer surface and the liquid surface. If solely free convection is the mode of heat transfer in the liquid the resistance can be estimated by relations similar to those for free convection. A treatment by Jacob et al. gives:

29

Nu = C . (Gr . Pr)n 7-25

Where C = 0,61; n = ¼ for vertical walls C = 0,16; n = 1/3 for a horizontal surface. Expressing the temperature difference in the Grashofs number as ∆t = (q/(A . h) the result becomes: h = const . (q/(A . H))(1/5) for a vertical wall (H = characteristic length) h = const . (q/A)(1/4) for horizontal surfaces.

NUCLEATE BOILING. For engineering purpose the most important region observed in the “boiling curve” is nucleate boiling. Different relations have been proposed to correlate the heat transfer in nucleate pool boiling of a saturated liquid.

The following relation is proposed by Hirschberg7:

)3/1()3/1(

2

24

Pr)/(')/(0185,0 ⋅⎥⎦

⎤⎢⎣

⎡⋅⋅

⋅⋅⋅∂∂⋅=

⋅=

μσρ

kAqbTp

kbhNu

where )"'( ρρ

σ−⋅

=g

b = Laplace’s constant [m]

Tp ∂∂ / = saturation pressure derivative [Pa/°C] ρ’ and ρ” = density of liquid and vapor [kg/m3] σ = surface tension of saturated liquid [N/m] k = liquid thermal conductivity of saturated liquid [W/(m.°C)] μ = dynamic viscosity of saturated liquid [N s/m2] q/A = heat flux density [W/m2] Pr = Prandtl number of saturated liquid A relation for nuclear boiling often cited is the following, given by Cooper8:

67,055,05,0

)(10log*2,012,0

))(10log(55 q

pMpCh

r

Rpr

⋅−⋅

⋅⋅=−

Where C = constant. For horizontal plane surface: C = 1 For horizontal (copper) cylinders: C = 1,7

pr = reduced pressure, = p/pcrit (pcrit is the pressure in the critical point) Rp = surface roughness expressed in μ m (!!) If no information available it is

often recommended to use Rp = 1. M = molecular weight q = heat flux [W/m2]

The results from the two relations can be represented in a similar way, simplified to: 3/2)/( AqAh k ⋅= or 23 TAh k Δ⋅= where Ak is a factor that can be estimated from the two given relations ∆T = the temperature difference [°C]

In the Appendix Heat transfer in Nuclear Boiling the factor Ak is given, estimated by the two correlations for a number of fluids. As seen the results differs somewhat, but tendencies are similar. 7 Hirschberg, H.G. Zeitschrift für Kältetechnik, Vol 19, p 155, 1966. 8 Cooper, M.G., “Heat Flow Rates in Saturated Nucleate Pool Boiling…” Advances in Heat Transfer, Vol 16. Academic Press, Orlando Florida.

30

HEAT TRANSFER IN FORCED BOILING INSIDE TUBES

Boiling inside channels has been the subject for a large number of research projects in different applications. The work by Pierre9, dealing with boiling of refrigerants in tubes, is one of the most cited and used. It is simple to use and in spite of this it gives normally quite good agreement between theory and experiments. In dimensionless form the relations are:

Nu = f(Re; Kf)

where kdhNu ⋅

= = Nusselt number; μπμ ⋅⋅

⋅=⋅=

dmd

Am && 4Re = Reynolds number

Lg

ΔhK⋅

=f = Pierre boiling number

k and μ = thermal conductivity and dynamic viscosity of saturated liquid ∆h = enthalpy change of the fluid along the tube [J/kg] L = tube length [m] m = mass flow rate in the tube [kg/s] q/A = heat flux (where A = π .d .L) [W/m2] g = gravitational constant (g = 9,81 m/s2) For incomplete evaporation (vapor quality x < 1 at the tube exit) Pierre gives10:

5,0Re0011,0 KfNu ⋅⋅= For complete evaporation, with 5 to 7 K vapor superheat at tube exit, the following relation is given by Pierre for the average heat transfer coefficient along the tube:

4,08,0Re01,0 KfNu ⋅⋅=

Notice: Since Re is proportional to mass flow, m, which is proportional to the heat flux (q/A), it follows that the temperature difference ∆T = (q/A)/h : For incomplete evaporation: ∆T is independent of the heat flux ! For complete evaporation: ∆T is proportional to (q/A)0,2 THE PRESSURE DROP IN FORECED BOILING INSIDE TUBES can according to the treatment by Pierre11 be estimated by:

mtm

mm v

Amn

xxx

dLfp ⋅⎟

⎠⎞

⎜⎝⎛⋅⎥⎦

⎤⎢⎣⎡ ⋅

+−

+⋅=Δ2

21

2&ζ

where ∆p = pressure drop [Pa] in a horizontal tube, length L and diameter d x1 and x2 = vapor quality at inlet and outlet xm for complete evaporation: 5,025,0 /4,4 Ldxm ⋅≅ for incomplete evaporation: 2/)( 21 xxxm +≅ vm = =(1-xm) .v’+xm

.v” = specific volume of vapor at x = xm tmζ = pressure drop factor for tube bends n = number of tube bends m = mass flow rate through the tube, A = cross sectional area ( = π . d2/4) fm = friction factor (a function of Re and Kf). For rough estimates can fm =

0,015 often be used.

9 Pierre, B., ”Värmeövergång vid kokande köldmedier i horisontella rör”, Kylteknisk Tidskrift, no 5, 1969. 10 Recent more accurate data of thermophysical properties for refrigerants suggests that the constants in the two equations should be multiplied by factor approximately 0,85. 11 Pierre, B., ”Strömningsmotstånd vid kokande köldmedier”, Kylteknisk Tidskrift, no 6, 1957. Also published in ASHRAE Journal, September and October 1964.

31

HEAT TRANSFER BY RADIATION Heat is transferred not only by conduction and convection but also by radiation. This is a completely different mode compared to the previous modes of heat transfer. In radiation heat is transferred by a similar process as visible light, but energy transfer is primarily in the infrared spectrum with wavelength between 0,8 and 400 μm (1 μm = 1.10-6 m = 0,001 mm = 10000 Ångström). The “monochromatic emissive power” (or the emissive power λE of a given wavelength, λ ) is obtained by the Planck’s law of radiation. For a perfect radiating body (a “black” body) the emissive power ( λbE ) is:

( )[ ]1)/(exp 25

1

−⋅⋅=

TCCEb

λλλ 8-12

where λ is the wavelength [m] ) 107403.3( 2 2162

1 mWchC ⋅⋅=⋅⋅⋅= −π ) 104387.1( / 2

2 KmkchC ⋅⋅=⋅= − (h = Planck constant; c = velocity of light in vacuum; k = Boltzmanns constant) Black body radiation has its maximum emissive power for a certain wavelength, maxλ , the value of which is dependent on the temperature, T, of the body. At higher temperature the maximum occurs at shorter wavelength. This is described by Wien’s displacement law:

T

3

max10898,2 −⋅

=λ [m] 8-13

where T is the absolute temperature of the emitting surface [K] By integration of the Planck’s law over the entire wavelength spectrum the total emissive power of a black body, Eb, is derived. The result is the Stephan-Boltzmann’s law: 4TEb ⋅= σ 8-3 where σ is the Stephan-Boltzmann’s constant = )15/( 4

214 CC ⋅⋅π = 5,669.10-8 [W/m2.K4]

Sometimes it is more convenient to write the Stephan-Boltzmann’s law on the following form: 4)100/(TCE sb ⋅= where Cs =σ .108 = 5,669

The emitted power from a real surface (not a perfectly black body) may be written:

4TEE b ⋅⋅=⋅= σεε 8-7

where ε is the (total) emissivity of the surface A surface having the same emissivity for all wavelengths is called a “grey” surface. Radiation is emitted from a body as discussed, but a body also absorbs incoming radiation. The absorptivity, α , describes how much of the incoming radiation energy that is absorbed by the surface (the rest is reflected, or if the body is transparent it is transmitted through the body without affecting it). From the law by Kirchhoff it is found that the absorptivity and emissivity must be equal for radiation at the same temperature

εα = 8-8 Some data for the emissivity (absorptivity) of different types of surfaces and different temperatures are given in Tab. A-10 in Holman.

32

The emissivity of a real surface is also a function of the direction. For surfaces where the emissivity is independent of the direction the radiation intensity follows the “Lambert’s cosine law” which states that the radiation intensity depends only on the projected area. HEAT TRANSFER BY RADIATION BETWEEN BLACK SURFACES The net heat exchange by radiation between two black surfaces, 1 and 2, depends on how they “see” each other. The following equation can be derived: )( 4

24

111212 TTAFq −⋅⋅⋅= σ

where ∫ ∫ ⋅⋅

⋅⋅⋅=1

2

21

2

211

12 coscos1

A A rdAdA

AF

πϕϕ =“view factor” 8-21

r = distance between the radiating areas ϕ1 and ϕ2 = the angles by which surface elements are exposed to each other. In the Appendix “View factors for radiation” data are given for two- and three dimensional geometries. In Holman Fig 8-12 to 8-16 also give “shape factors” (equal to view factors) for several geometries. More than two surfaces can be involved in radiation exchange. For instance we can define view factors between three surfaces 1, 2 and 3 where F12 is the view factor between surfaces 1 and 2; F1(2+3) is between surface 1 and surfaces (2+3) and F13 for radiation between 1 and 3. The following simple rule holds:

F12 = F1(2+3) – F13

HEAT TRANSFER BY RADIATION BETWEEN GREY SURFACES Radiation exchange can also be treated by introducing radiation resistances, Rrad: Consider the blackbody emissive power as the driving potential

q12 = (Eb1 -Eb2) / Rrad = σ ⋅ (T14 -T2

4) / Rrad The radiation resistance for grey surfaces is the sum of surface and space resistances

Rrad = Rsurface1 + Rspace + Rsurface2

It can be shown that Rsurface = (1-ε)/(ε ⋅ A) and Rspace = 1/(A1⋅F12)

For heat exchange between two bodies, this approach gives:

22

2

12111

1

42

41

21

212112 111

)(

AFAA

TTRRR

EER

EEqsurfacespacesurface

bb

rad

bb

⋅−

+⋅

+⋅

−−⋅

=++

−=

−=

εε

εε

σ

Special cases: Two surfaces where one (concave) area A1 is totally enclosed in another (A2), then F12= 1:

qA

T TAA

1

14

24

1

1

2 2

1 11

=⋅ −

+ ⋅ −

σ

ε ε

( )

( )

and if A1 << A2 q/A1 = ε1⋅σ⋅(T14 - T2

4)

33

Two parallel infinite plates: F12 = 1, A1 =A2 qA

T T=

⋅ −

+ −

σ

ε ε

( )14

24

1 2

1 11

For three (or more) body problems: A radiation network can be drawn.

The potentials between Rsurface and Rspace is called Radiosity, denoted J. The Radiosity J is the total radiation leaving the surface (emitted and reflected incoming). The procedure is as follows: • Calculate all resistances. • For the nodes Jn, the sum of the energy

flows into each of the nodes must be zero, e.g.: E J

A

J JA F

J JA F

b1 1

1

1 1

2 1

1 12

3 1

1 1310

−−⋅

⎛⎝⎜

⎞⎠⎟

+−⋅

+−⋅

=

⋅

εε

.

• This will result in a system of equations for Jn which can be solved to give all Jn.

• Calculate the heat flows from each surface! RADIATION HEAT TRANSFER COEFFICIENT Heat exchange by radiation can also be expressed by means of a “radiation heat transfer coefficient”, hRad. In this concept the heat transfer, qRad, between two surfaces is expressed: )( 211 TTAhq RadRad −⋅⋅= If this equation is identified by the basic relation for radiation heat exchange between black surfaces it is obvious that the radiation heat transfer coefficient can be expressed by:

21

42

41

12TTTTFhRad

−−

⋅⋅= σ

Radiation between two black bodies which “sees” each other perfectly (F12 = 1) will have a radiation heat transfer coefficient, hRadBlack, as given in the following table:

t1 t2, °C -50 -10 0 10 20 50 100 200 400 800 1000 -50°C 2,52 3,28 3,50 3,73 3,97 4,78 6,39 10,8 25,6 88,3 141,7 -10 4,13 4,37 4,63 4,89 5,77 7,52 12,2 27,7 92,5 147,2 0 4,62 4,88 5,15 6,05 7,84 12,6 28,3 93,6 148,6 10 5,15 5,43 6,35 8,16 13,0 28,9 94,7 150,1 20 5,71 6,65 8,51 13,5 29,5 95,9 151,6 50 7,65 9,62 14,8 31,5 99,4 156,1 100 11,78 17,4 35,1 105,8 164,3 200 24,0 44,0 120,6 182,6 400 69,2 158,9 228,8 800 280,3 368,8 1000 468,0

As seen radiation heat transfer is rapidly increasing at higher temperatures!

The concept of radiation heat transfer coefficient is convenient to use in many engineering situations. It may be noticed that at room temperature hRadBlack is of the same order of magnitude as the coefficient of heat transfer in free convection in room air.

35

APPENDIX Heat Exchanger Effectiveness Counter current 37 Co-current 38 Cross current 39

Fin Efficiency 40 Shape factors for heat conduction 41 – 43 Temperatures in transient conduction Slab 44 Cylinder 45 Heat transfer in falling films 46 Heat transfer in Nuclear Boiling 47 View factors for radiation Two dimensional 48 - 49

Three dimensional 50

37

Heat Exchanger Effectiveness: Counter current

Source: B. Pierre: Mek värmeteori, forts kurs, KTH, Stockholm 1982

38

Heat Exchanger Effectiveness: Co-current


39

Heat Exchanger Effectiveness: Cross current


40

Fin Efficiency

Source: E. Granryd: Tillämpad termodynamik, KTH, Stockholm 1994

41

Shape factors for heat conduction

Source: E. Hahne, U Grigull: Int J. Heat and Mass transfer, Vol 18, 1975, p 751-767

42

Shape factors for heat conduction, continued


43

Shape factors for heat conduction, continued.


44

Temperatures in transient conduction – Plate


45

Temperatures in transient conduction – Cylinder


46

Heat transfer in falling films Laminar film

Heat transfer in falling films Turbulent film


47

Heat transfer in Nuclear Boiling Two examples

Hirschberg

0,5 1 10 401

2

5

10

20

Pressure, bar

Ak

[1/K

*(W

/m2)

^(1/

3)]

R134aR134a

WaterWater

AmmoniaAmmonia

PropanePropaneR152aR152a

EthaneEthaneEthyleneEthylene

IsoButaneIsoButane

CO2CO2

Hirschberg relation h = Ak*q(2/3)

Cooper Results for horizontal plane surface

0,5 1 10 401

2

5

10

20

C02C02

WaterWater

Cooper relation h = Ak*q(2/3)

R134aR134a

IsoButaneIsoButane

Ak

[1/K

*(W

/m2)

^(1/

3)]

Pressure, bar

PropanePropane

EthyleneEthyleneEthaneEthane

AmmoniaAmmonia

R152aR152a

48

View factors for radiation – Two dimensional geometries

Source: F. P. Incropera, D. P. De Witt: Fundamentals of Heat and Mass Transfer, 3rd ed., New York, 1990. Reference: Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982.

49

View factors for radiation – Two dimensional geometries

Source: F. P. Incropera, D. P. De Witt: Fundamentals of Heat and Mass Transfer, 3rd ed., New York, 1990. Reference: Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982.

50

View factors for radiation – Three dimensional geometries

Source: F. P. Incropera, D. P. De Witt: Fundamentals of Heat and Mass Transfer, 3rd ed,

New York, 1990. Reference: Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill, New York, 1982.

525

t, på - diagram förfuktig luft

på påx

t

t

t

h

h

h

ϕ

t

frostkwd )/( αα )/( kwd αα

Eric Granryd

Fig 12.13b t,på-diagram för fuktig luft.

bpalm

Text Box

temp, vapor pressure diagram for moist air

bpalm

Text Box

Vapor contents, kg H2O per kg dry air

bpalm

Text Box

Icy surface t<0°C wet surface t>0°C

bpalm

Text Box

wet surface

bpalm

Text Box

icy surface

bpalm

Text Box

t, p-diagram for moist air

Reviewer

Text Box

51

THERMAL PROPERTIES Tables of thermal and physical data for a selection of Solid materials, Liquids and Gases. Appendix to Summary of formulas in Heat Transfer Content: TABLE 1. Selection of Metals TABLE 2. Selection of Non metallic materials TABLE 3. Selection of Building materials TABLE 4a. Dry air TABLE 4b. Humid air TABLE 5a. Water: saturated liquid and vapor at low temperatures TABLE 5b. Water: ice, liquid and vapor TABLE 6a. Selection of liquids TABLE 6b. Selection of aqueous solutions TABLE 7. Selection of gases TABLE 8. Selection of refrigerants TABLE 9. Selection of substances (gases)

TABLE 10. Heat resistances due to fouling in heat exchangers UNITS AND CONVERSION FACTORS

52

TABLE 1.

53

TABLE 1 (continued).

54

TABLE 2.

55

TABLE 3.

56


57

TABLE 4a.

58

TABLE 4b Humid air at (atmospheric) pressure, p = 1,000 bar. Enthalpies are set to zero (h=0) for liquid water and air at 0°C. Water content and enthalpy expressed per kg dry air.

Source: Bäckström, M. Ahlqvist, D.: Isoleringar av kyl- och frysutrymmen, IsoleringsAB WMB, Stockholm 1963.

TABLE 5a.

59

TABLE 5b.

60

TABLE 6.

61


62


63

TABLE 7.

64


65

TABLE 8.

66


67


68


69

TABLE 9.

70


71


73

TABLE 10

74

UNITS AND CONVERSION FACTORS

75

UNITS AND CONVERSION FACTORS (Continued)

heat transfer collection of formulas and tables_ 2009

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