HTR LectureNotes 2015 Summerterm

Mechanical Engineering Faculty/ Design and Development

SUMMER TERM 2015 PROF. DR.-ING. U. SCHELLING

Hochschule Konstanz Technik Wirtschaft und Gestaltung

HEAT AND MASS TRANSFER

Contents Heat and Mass Transfer

1 Introduction .......................................................................................... 1.1 Function of Heat Transfer ...................................................................... 1.2 Design of Heat Transfer Devices ........................................................... 1.3 Effects of Heat Transfer ......................................................................... 1.4 Further Subjects in this Course of Lectures........................................... 1.5 Mathematical Fundamentals .................................................................. 2 Heat Conduction................................................................................... 2.1 Fundamental Principles of Heat Conduction.......................................... 2.2 Steady State Heat Conduction............................................................... 2.3 2D-Heat Conduction............................................................................... 2.4 Nonsteady State Heat Conduction......................................................... 2.5 Contact Temperature ............................................................................. 3 Convective Heat Transfer.................................................................... 3.1 Heat Transfer and Influencing Parameters ............................................ 3.2 Heat Transfer with Forced Convection................................................... 3.3 Heat Transfer with Free Convection ...................................................... 3.4 Heat Transfer During Phase Change..................................................... 4 Heat Transmission ............................................................................... 4.1 Heat Transmission Laws........................................................................ 4.2 Intermediate Temperatures.................................................................... 4.3 Fouling Resistance................................................................................. 5 Heat Exchanger .................................................................................... 5.1 Parallel and Reverse Flow Heat Exchangers ........................................ 5.2 Number of Transfer Units (NTU) ............................................................ 6 Special Problems in Heat Transfer..................................................... 7 Radiation ............................................................................................... 7.1 Concepts and Basic Equations .............................................................. 7.2 Radiation Exchange Between Solids ..................................................... 7.3 Gas Radiation......................................................................................... 8 Moist Air ................................................................................................ 8.1 Material Data and Definitions ................................................................. 8.2 State Variables ....................................................................................... 8.3 Phase Diagram for Moist Air .................................................................. 8.4 Changes in State in the Case of Moist Air ............................................. 8.5 Gas-Vapor Mixtures at p 1 bar ............................................................ 9 Mass Transfer ....................................................................................... 9.1 Diffusion ................................................................................................. 9.2 Mass Transfer ........................................................................................

Heat and Mass Transfer 1-1 Schelling / HTWG Konstanz

Schelling Copying permission only for internal use 1-HTR-Chap01-V08-2015.03.08.docx

1 Introduction

1.1 Function of Heat Transfer In courses of lectures on basic thermodynamics, it is taught that heat and work are process parameters, whereby, the supply or dissipation of them causes corresponding changes to the state of the system. For the process parameter, work, appropriate formulas are taught (changing volume work, displacement work, ...) or is already assumed (lifting work, electrical work, ...) so that, already in this respect, the resolution of more complex tasks was possible. Only numerical values were specified for quantities of heat or flows of heat, or, by the assignment of tasks, such values were determined, without exploiting them further. The reason lies in the multitude and complexity of the effects and formulas, which require their own course of lectures The Assignment of Heat Transfer:

1.2 Design of Heat Transfer Devices1 There is a multitude of different designs of devices for heat transfer, which can be differentiated and classified according to many criteria. The most important differentiation results from the principle of heat transfer ( regenerator or recuperator) and on the type of flow ( parallel, reverse flow, cross flow)

1 The previously used expression, heat exchanger (also occasionally used today) is avoided, because heat cannot

be exchanged between hot and cold, but can only be transferred from hot to cold. The special case of heat transfer

by radiation will be discussed later.



Switching Type Closed Circuit Type Figure 1.1: Basic Types of Regenerators (Figure: Merker/Eiglmeier) Differentiated according to the type of flow: Figure 1.2: Temperatures with Parallel Flow, Reverse Flow & Cross Flow HTD Differentiation by the Type of Design:

Tube bundle HTD

Plate- HTD

Spiral Heat Transfer Device

Figure 1.3: Designs of Heat Transfer Devices (Figure: Merker/Eiglmeier)

Parallel Flow HTD Reverse Flow HTD Cross Flow HTD

Hot Flow Cold Flow

A = Aus(tritt) = out(let) E = Ein(tritt) = in(let)





1.3 Effects of Heat Transfer Heat transfer is a form of energy transfer and therefore always related to material. It is, however, possible to differentiate between heat transfer with and without a transfer medium.

. Each of these effects depends on many parameters. However, it can always be described by a simple, fundamental formula with detailed problems being "placed" in a single coefficient. Later on, the accurate determination of the coefficients becomes the main task of the solution of heat engineering problems. BASIC FORMULAS: Thermal conduction: (1.1) Convection: (1.2) Radiation: (1.3)

Because several effects frequently influence heat transfer simultaneously, the combined effect is described as heat transmission.

Heat transmission: (1.4) In the above:

2

2

2 4

12

Q Heat Flow W

Thermal Conductivity W m K

Heat Transfer Coefficient W m K

k Heat Transmission Coefficient W m K

C Radiation Exchange Coefficient W m K

s Wall Thickness [m]

A Area [m]



In some circumstances, it is expedient and instrumental to introduce the concept of heat resistance. This concept is based on the concept of electrical resistance, with which there is an analogy and makes understanding it easier. Thermal resistance: (1.5) The analogy becomes evident immediately with a comparison to Ohms Law:

Temperature (difference) Electrical voltage (difference) Heat flow Electrical current Thermal resistance Ohm's resistance

1.4 Further Subjects in This Course of Lectures For convection, the transfer of heat is very much influenced by the magnitude and type of flow. This is also the origin of the expression "Thermo-fluids", the name given earlier to this course of lectures. Correspondingly, the chapter Convective Heat Transfer resorts to the experience and results of Fluid Mechanics. In this context, the fundamental equations of the mechanics of thermo-fluids are clearly summarized here. The values searched for, in an area with a flow through it,

w

= the velocity vector of the flow with the components u,v,w

p = the static pressure in the flow

= the density of the fluid (gas or liquid)

T, = the temperature in K or C are a function of the coordinates of the area and of time t. To calculate these values, the balance equations are available as follows: Fundamental Equations of Thermo-Fluid Mechanics:

The status equations, functionally links the (scalar) state variables of pressure p, density and temperature T to one another.



In addition to dealing only with heat transfer problems, at the end of this course of lectures, two chapters will be addressed that are indirectly connected with the transfer of heat: Moist Air: Special case of the general problem of the transfer of material.

The thermodynamic behavior of multi-material systems (here: air/water) will be addressed, combined with phase changes (mainly: liquid/gaseous condensate) which may be associated with it.

Diffusion: General instance, in which one, or more materials, move relative to one another

with no appreciable macroscopic flows.

1.5 Mathematical Fundamentals Within the scope of the course of lectures, the following mathematical concepts are occasionally required and applied to the parameters of interest for flow and heat transfer: A) Concepts:

Scalar: Non-directional physical parameters, e.g., T, p,

Scalar field: Assigning scalar parameters to points, lines or surfaces in an

area, e.g., the temperature distribution in an area Contour lines Locations of equal scalar magnitude, such as isotherm, isobars or areas: Scalar product: (So-called internal product, also refer to the course of

mathematical lectures)

cos2121 vvvv

1v

= Vector with the component zyx vvv 111 ,,

2v

= Vector with the component zyx vvv 222 ,,

zzyyxx vvvvvvvv 21212121

Vector: Physical quantity, dependent on direction, e.g., velocity,

acceleration, force Vector field: Assigning vector magnitude to points, lines and surfaces in an

area; e.g., velocity field, force field Vector product : Also refer to the course of mathematical lectures

bac

with the properties:

c

perpendicular to a

and b

a

, b

and c

form a right-handed system of coordinates

length of the vector: sin bac



B) Integration:

So-called "Engineering Integral":

dxxxF

x

n *

0

*)(

Integral of a sum:

dxwdxvdxudxwvu )(

Substitution method where )(tx :

dtttfdxxfxF

tx

)()]([)()(**

00

*

C) Derivative:

Derivative dxdyy /' of the function nx

Derivative of products:

uvvuvu '')'(

Derivative of quotients:

2/)''()'/( vuvvuvu

Chain rule with: )(),(),( xttuufy

)()()( xtufdx

dy

Derivative of a function with multiple variables ),,,( tzyxfu

with respect to one variable (= partial differential), e.g. with respect to x

x

u

with respect to all variables (= complete or total differential)

dtt

udz

z

udy

y

udx

x

udu

D) Operators: Operators are names of arithmetic instructions for differential computing directive, that is, a type of shorthand for longer mathematical operations in the differential range. A disadvantage of the operator mode of writing is the compulsory loss of clarity, because of the greater abstraction. The advantage is the generally valid notation, which is short as well as independent1, of the coordinate system selected.

1

The associated computing directives are, admittedly, dependant on the coordinate system, however, they are

tabulated in many textbooks of fluid mechanics. The examples that follow always relate to the Cartesian coordinate

system.



The gradient of a scalar:

grad

may be any scalar function of, for example, x, y, z and t.

The change in any scalar function on progressing about the path element ),,( dzdydxsd

is,

on the one hand, a result of the total differential:

dzz

dyy

dxx

d

At the same time, this total differential can also be considered formally as a scalar product of

two vectors, of the vector grad and of the path element sd

:

d , , dx,dy,dzx y z

Hence, for example, the specific heat flow (refer to Chapter 2) dAQdq can be represented without coordinates:

Tgradq

The vector, grad , is perpendicular to the contour line and indicates the direction of the

greatest change in . In other words, heat flows automatically in the direction of the largest

temperature change (= temperature gradient). The Nabla Operator (it is a symbolic vector):

, ,x y z

The Laplace Operator:

2

2

2

2

2

2

zyx

Example: The "divergence of the velocity" may be formally interpreted as a scalar product of the

Nabla Operator , with the velocity vector, w

; However, it actually only represents a calculation

specification for the components of the vector, ),,( wvuw

.

div w =


Schelling Copying permission only for internal use

2 Heat Conduction

2.1 Fundamental Principles of Heat Conduction Heat conduction is the transport of energy resulting from molecular interaction in solid, liquid or gaseous media under the influence of a temperature gradient.

i) Steady State Heat Conduction The description is based on the empirical Fouriers Law1, which was established in 1822 and has consistently, without exception, been confirmed by measurements.

The temperature drop, inside and outside, at the wall is described by the heat transfer

coefficient (refer to Chapter 3). If the two temperatures T1 and T2 are kept constant, then the amount of heat flowing through the wall is Q: (2.1-1)

1 J.B. Fourier, French Physicist (1768-1830)

warm

body at rest

cold

body at rest

body at rest

Heat Insulation

warm cold

x

T

How long does it take for the temperatures to equalize?

How much must I heat so that the temperatures do not change?

How large is the temperature difference at a fixed heat flow?



where A = area of the wall [m]

= thermal conductivity of the wall material [W/m K] s = wall thickness [m] t = time [s] The amount of heat, flowing in a unit of time, is given by:

t

QQ (2.1-2)

The heat flow density or the specific heat flow that flows through each square meter of wall surface is (2.1-3)

In an arbitrarily thin layer dx within the wall, the temperature drops by dT ; accordingly the heat flow density there is (2.1-4) whereby the minus sign must be introduced, so that the heat flows physically correctly from hot to cold. This is the Fourier Thermal Conductivity Law, in differential notation.

The proportionality factor, , depends largely on the material and varies by several powers of ten.

Figure 2.1: Thermal Conductivity of Different Groups of Materials at Room Temperature (according to Cengal)



Figure 2.1: Thermal Conductivity of Different Media as a Function of Temperature VDI-WA, Dea 1 Source: Bosnjakovic I according to Wagner/Vogel-Verlag

The thermal conductivity, , is usually a function of the temperature (and, with gases, also of the pressure. Frequently, however, it can be assumed to be constant within ranges. If this is not possible, because of high dependence on temperature or large changes in

temperature, equation (2.1-3) must be modified. Where )( f , it follows from the equation

(2.1-4) that Separation of the variables1 and integration:

After the average thermal conductivity, 2,1 , is introduced in the range between 1 and 2

where (2.1-5) the equation for the heat flow density is: (2.1-6)

1 The heat flow is assumed to be constant, because a one-dimensional case (the flat wall) is considered here.



ii) Multidimensional Heat Conduction Until now, the process of one-dimensional heat conduction was always considered in Cartesian coordinates. What happens in the two dimensional or three dimensional case, or if the material behaves anisotropic1?

In the general view, the heat flow becomes a heat flow vector, which, in Cartesian coordinates, reads:

1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

q T x

q T x

q T x

(2.1-7)

The frequently occurring case of the multidimensional heat conduction with isotropic conductivity, that is, with the same material behavior in all spatial directions it can be written more simple, and is generally referred to as a Fourier statement. (2.1-8) For handling technical tasks, a one-dimensional calculation is frequently possible as an approximation and, as a rule, it can be assumed that the thermal conductivity is constant within ranges, so that the simple equation (2.1-6) can be used.

1 the properties are dependent from direction in space. e.g. with filamentary materials like wood

warm body cold

body

body at rest

warm

cold

cold

cold

thermal insulation



iii) Unsteady State Heat Conduction For the derivation, a balance is set up at a (one-dimensional) disk volume element, including a

heat source with power density w* m

W

First Law of Thermodynamics (steady, closed system):

pot. & kin. energy

12 12 2 1 0 0Q W U U specific:

supplay and removal of energy temperature change

*

12 12 2 1 2 1 vq A V w m u u m c T T (2.1-9)

mass specific parameter volume specific parameter area specific parameter differential at the volume element in the time section

*x x dxA q q A dx w dm c T A dx c Tt t

Taylor series development for heat flow (only to the 1st order)

...xx dx xq

q q dxx

*

xq dx dx w dx c Tx t

*xq

c T wt x

With Fourier statement (2.1-8) xT

qx



*T

c T wt x x

With constant material data, it follows that:

*

T T w

t c x c

*

T T wa

t x c

(2.1-10 / only cart. coord.)

* *

T w w

a T a Tt c c

(2.1-11 / generally valid)

The newly introduced thermal diffusivity a is only a material parameter.

ma

c s (2.1-12)

With no heat source:



2.2 Steady State Heat Conduction Under steady state conditions and with no internal source of heat, equation 2.1-11 simplifies to (2.2-1) In Cartesian coordinates, this is: Example: Two steel or Plexiglass bodies of identical shape

2.2.1 Heat Conduction through a Wall, Cylinder, Hollow Sphere 2.2.1.1 Heat Conduction through a Flat Wall Commencing from the Fourier statement

dx

dTA

dx

dTAqAQ

i

ii

one-dimensional

Steel Plexiglass



after integration 2 1 1 2

2 1

T T T TQ A A

x x s

for a multilayer wall:

1

21112

s

TTAQ

2

32223

s

TTAQ

. . .

n

nnn

s

TTAQ 1

By elimination of temperatures in series (or by transposition from T1-T2 or T2-T3, adding equations )

1 2

1 2

1 1

...n

n

n

s s s

A T TQ

(2.2-2)

2.2.1.2 Heat Conduction Through a Hollow Cylinder

( )dT

Q A rdr

where ( ) 2A r L d L r

dr

dTrLQ 2

Separating the variables then integrating: 2dr L

dTr Q

or 1 22

1

2

ln

LQ T T

r

r

(2.2-3)

Cylinder, r1 = internal radius

Caution: Q > 0 by loss Q < 0 if heat is applied

r

T

x



The so-called pipe factor, fR, is obtained by a comparison with a flat wall:

1 2 1 22

( ) ( )Wall R sA L

Q f T T T TQ

s ^= r2 r1

21

2 1

2 lnR r

r

r rf

r

pipe factor (=Rohr-factor) (2.2-4)

Example 2.1 Pipe diameter 2 m; thickness of the insulation 10 cm r1 = 1m r2 = 1,1m

9538,0ln1,1

1,0

1

1,1Rohrf 4,6% deviation

Example 2.2 Pipe diameter 10 cm; thickness of the insulation 10 cm

606826,0ln15

10

515Rohr

f 39% deviation

for pipes with thick walls, the flat wall method of calculation cannot be used, not even as

an approximation doubled insulation thickness does not halve the losses Multilayer Hollow Cylinder

Heat flow Q = constant through all layers

transpose equation 2.2-3 according to T record for each layer

T1 - T2 = + T2 - T3 = ------------------

T1 - Tn+1 =

1 1

32 1

1 21 2

2

1 1 1ln ln ... ln

n

n

n n

LQ T T

rr r

r r r

(2.2-5)



2.2.1.3 Heat Conduction Through a Hollow Sphere ^= insulated spherical container

- analog to a cylinder: ( )dT

Q A rdr

with 2( ) 4A r r

4 dT

Q rdr

Separation of the variables, integration:

2

2

1

1

4 1 4

rT

Tr

dr IntegrationdT T

r Q r Q

1 2

1 2

4

1 1Q T T

r r

(2.2-6)

Sphere Example 2.3 Sphere Di = 1 m; T1 = 100C (constant); Ta = 30C;

Mineral insulation where s1 = 10 cm; s2 = 20 cm; s3 =; = 0,04 KmW

Total: Heating necessary?

1 1 10,5 0,6

40,04 100 30 105,5Q W

2 1 10,5 0,7

40,04 100 30 61,56Q W

3 1 10,5

0

40,04 100 30 17,58Q W

a sphere always has a minimum heat flow. How large? (2.2-7) multilayer sphere same procedure as for a cylinder

1 1

1 1 2 2 2 3 1

4

1 1 1 1 1 1 1 1 1...

n

n n n

T TQ

r r r r r r

(2.2-8)



2.2.2 Thermal Resistance The analogy between electric current and heat flow applies Kirchhoff's Laws

Q

TR

W

K Thermal Resistance (2.2-9a)

T

QR

Q T Tq

A R A r

(2.2-9b)

ARr m K

W

specific thermal resistance (2.2-9c)

According to Kirchhoff, for a

- circuit in series: , 1 2 , ,1 ,2... ..el tot totR R R R R R

- circuit in parallel: , 1 2 , ,1 ,2

1 1 1 1 1 1... ...

el tot totR R R R R R

2.2.2.1 Thermal Conductivity of Material with Layers depending on the orientation of layers to heat flow layers perpendicular to heat flow layers longitudinal to heat flow For coefficient , the sequence and number of layers is not important formula can be derived at two layers. i) Heat Flow Perpendicular to the Layers series connection of resistances

a btot i

a a b b

s sR R

A A

General: TAs

Q Statement:

tot

Q A Ts

material a material b

Qperpendicular , perp.

material a material b

Qlongitudinal, long.



1 1tot tot tot totp

a b a btot

a b a b

s s s sQ

s s s sA T A R A

A A

General:

...

totp

a b n

a b n

s

s s s (2.2-10)

ii) Heat Flow Longitudinal to the Layers parallel connection of resistances

not Q const , but T const , resp. iQQ

where TAs

TAs

QQQ bb

ba

a

aba

a a b bQ s

A AT

statement l totQ s

AT

General: ...a a b b n n

l

tot

A A A

A

(2.2-11)

Note: l is always larger than p for insulating, always stack up transversely

2.2.2.2 Quasi-1D-Heat Conduction E.g. Wall of hollow building blocks Dual-shell container with insulation and support Extruded aluminum or plastic profiles 1 1 2 Representative section

Q Q Assumption: T = f(x) f(y) quasi one-dimensional! What is the adequate thermal resistance?

y

x



2.2.3 1D Heat Conduction with Heat Transfer Up to now: 3-D geometry (wall, cylinder, sphere) and nevertheless 1-D heat flow through - adiabatic boundary or -large expansion - symmetries (rotational, spherical) - simplification quasi-1D Up to now: Wall temperature T1 and Tn+1 specified not common Standard: Air (water, oil ) temperatures inside and outside known (or sought) convection must be included in the calculation Statement: Analogy to the electrical resistance

Equation 1.5 Q

TR

or

1 2F F

tot

T TTQ

R R

Equation 1.2 TAQ A

R

1

1

1

A

R

2

2

1

Equation 1.1 Q A Ts

1

1

1

1R

As

2

2

2

1R

As

...iR

R1 R1 R2 R2

T1 T2 T3 TF2 TF1



Flat Wall:

1 2

1 1 1

j

tot i

j j

sR R

A

W

K (2.2-12a)

Hollow Cylinder

1

1

1 1 1 1ln

2

n

j

tot

ji i j j a a

rR

l r r r

W

K (2.2-12b)

Hollow Sphere

2 2

1 1

1 1 1 1 1 1

4

n

tot

ji i j j j a a

Rr r r r

W

K (2.2-12c)

2.2.4 Heat Conduction in Cooled Ribs What should the length of a cooling rib be? (or heating rib) What should the length of a teaspoon be? Assumption: Slender rib approximately 1D solution

dx

dTAQx

)( TTAQ Wand

Heat Transfer (HT): ( ) ( )x xdQ A T T U dx T T (U=circumference)

Heat Conduction (HC): dx

dTAQx

(Fourier)

dxdx

QdQQ xxdxx

(Taylor series) (^= linear approximation)

Balance: x dx xQ Q dQ



= x xdQ

dx dQ U dx T Tdx

where xd dT

A U T Tdx dx

then divide by A ( const )

2

2

x x

d T UT T m T T

dx A

(2.2-13)

where the rib parameter: A

Um

(2.2-14)

General solution:

1 2mx mxT T C e C e

Boundary conditions: x=0 T = T0

x=h 0dT dx ^= Thermal conduction at head of rib neglected

or: x xT T

hmhm

xhmxhm

ee

ee

TT

TT

0

(2.2-15)

Derivation given in textbooks, e.g. Wagner.

Whith 12

cosh( ) ( )x xx e e it follows that

TT

TT

0

(2.2-16)

Figure 2.3: Functions for Heat Conduction in Cooled Ribs (Source: Wagner & Merker)



Rib end (x=h): 0

cosh(0)

cosh( )

hT T

T T m h

)cosh(

0

hm

TTTTh

(2.2-17)

Total heat flow Q : 0

0dx

dTAQQges

With follws

TThmmAQ 00 tanh (2.2-18)

Rib Efficiency:

Comparison with idealQ where TRib = T0 , that is with id

0 0

0

h

x

R

ideal

T T U dxQ

U h T TQ

(2.2-18)

R

02

0

tanh tanh1A m m h T T m m h

U h T T m h

tanh( )m h

m h

(2.2-19)

Optimum rib is where mh 12

(R has limited meaning! Consider the length of the rib)



2.3 2D-Heat Conduction In the general case of dual-dimensional heat conduction, the Fourier statement (Equation 2.1-8) must be integrated over the area of interest.

(Eq. 2.1-8) i

ix

Tq

or Tgradq

With the general running coordinates1, the heat flow is calculated from one isothermal surface, A1, to a second isothermal surface, A2, as follows:

21

2211

AA

dAnTdAnTQ

For constant material data, the equation can be simplified by a so-called Form coefficient S, which includes all geometry-dependent terms.

21 TTSQ (2.3-1) where

12

22

12

11

21

TT

dAnT

TT

dAnT

SAA

(2.3-2)

By reference to a characteristic length (e.g. length of the pipe) the dimensionless form coefficient is obtained

lSSl / (2.3-3)

and the heat flow per unit length

21 TTSlQ l (2.3-4) Form factors for plates, tapes, straight ribs, pipes, spheres, disks, etc. are tabulated in the VDI-Wrmeatlas, Kap Ea,.

1 The coordinate, n, is always defined perpendicular to an isothermal surface; heat therefore always flow in the

direction of n



out inm m

2.4 Unsteady State Heat Conduction The equation, for the unsteady state heat conduction with a heat source, is derived for the one-dimensional case in Chapter 2.1 and then generalized for any number of dimensions

(Eq. 2.1-10) c

W

x

Ta

t

T

*

2

2

or: (Eq. 2.1-11) c

WTa

t

T

*2

This equation can be interpreted in words; from the energy balance, it follows that:

The heat capacity affects the time-space distribution of the heat flow. Conceptually, this can be represented by the following comparison: Accordingly, this equation is a variation of Equation 2.1-11, shortened by the source term and, for the reasons of systematy, is derived new in this chapter: (2.4-1) where (2.4-2) The solution to this differential equation (2.4-1) depends on the starting and boundary conditions. Possible boundary conditions for the surface are (Index 0):



First type: T0 = Tenvironment = constant (constant temperature; )

Second type:

0

0

qn

T constant (constant heat flow, e.g. electric or nuclear)

Third type: tenvironmenTTn

T

0 General case

Simple geometry (plate, cylinder, ) and simple boundary conditions analytical solution possible

General case only numerical solution possible, e.g. if = f(T)

Important parameters are: t, a, , , characteristic length L dimensionless parameters

dimensionless temperature A

T T

T T

(2.4-3)

A=Anfangs = Initial Temperature

dimensionless time 2L

taFo

(2.4-4)

Ratio (2.4-5)

where = thermal conductivity of the solid

( , )f Bi Fo standardized representations possible

2.4.1 Specified Wall Temperature Model representation: Tape passes through a bath Assumption: very large Tsurface = T0 = constant HT at the sides neglected HC in longitudinal direction neglected

Heat conduct.-resist.inside Heat transfer-resist.outside.

1D problem

1~

L LBi

Bath with T = T0 = constant



(A) At start of tape

If the time is short (or the tape is thick) the middle temperature initially remains unchanged Where flow coordinates y = s x (2.4-6)

(B) At end of tape

Temperature in the middle of the tape clearly differs from TA.

The differential equation 2

2

y

Ta

t

T

(=2.4-1) has the following asymptotic solutions1:

Solution:

A) For short times (at the start of the tape)

Time t short or argument of the function large

ta

yerf

ta

xserf

TT

TT

A 220

0 (2.4-7a)

Error function

where the characteristic dimension L = y = s x

yFoy

ta

L

taFo

22 local Fo - number

yFoerf

2

1 (2.4-7b)

1 for the derivation, refer to literature, e.g. Baehr or Merker-Eiglmeier

y

T



that is, the physically complex problem (function of TA, T0, , , cp, geometry) is reduced to the mathematical solution of the erf - function (Gaussian error integral)

2

0

2 zerf e dz

(2.4-8)

Refer to mathematical tables or data sheet HB-F3

B) For long times (at end of tape)

The temperature profiles are similar

Statement: ( ) ( )T f y g t

Solution:

0

0

2

244 cos2A

a t

sT T x eT T s

(2.4-9a)

or. (2.4-9b) Example:

A plastic tape at 20C, where a = 0.1 x 10-6 m/s, goes into a bath with boiling water. When will the center of the 6 mm thick tape reach a temperature of 60C?

Assumption: Surface of tape = T0 = 100C immediately Center of tape (2.4-9c)

2

0,9



c) Medium Times

Temperature field can be determined only by a series development with a numerical solution Solution in a dimensionless form, considering the outer heat transfer at the same time

2.4.2 Unsteady State Heat Conduction With Convection

Temperature field T = T(t,y) for plate (and cylinder or sphere) results in a Fourier series with an infinite number of terms

Generally, only the first or second terms are relevant (engineering accuracy: if the data for the material is accurate only to % )

Main interest for:

heat flow in or from the body

surface temperature

middle temperature

caloric average temperature Heat Flow Konvektion

temperature gradient thermal stress

Area A

2

0

ydy

dTAQ

Derivation of the Fourier series where y = 0 (refer to literature e.g. Wagner)

Convection



Solution:

2 2

24

1

4( )

n a t

sA

n

Q t A T T es

Essentially of interest for the quantity of heat already transferred

0

( )

t

Q Q t dt initial enthalpy difference time function

2 2

4 2

2 21

8 12 1

n a t

s

n

t A

n

Q A s c T T en

V

simplified:

( ) A QQ t V c T T f (2.4-10)

with fQ, refer to Help Sheet D-05 % Factor of the maximum quantity of heat transferred For cylinders and spheres, refer to Help Sheet D-06 and Help Sheet D-07, with the Caloric average temperature, defined as:

V V

V

c T dV T dV

TVc dV

(2.4-11a)

(approximately for = constant, c = constant) and, with that, dimensionless:

A

T T

T T

(2.4-11b)



2.5 Contact Temperature If two bodies with different temperatures come into contact with each other, on the contact surface, the so-called contact temperature TK sets in immediately. In case of infinitely expanded bodies, the temperature distribution in each of the bodies corresponds with the course of the temperature of a semi-infinite body with erratic changes in the surface temperature and can therefore be calculated using the Gaussian error function as per Chapter 2.4.1. If one is only interested in the contact temperature of the two bodies, then the solution of the problem is reduced to the determination of the heat penetration coefficients b1 and b2 of the affected bodies (derivation see literature, e.g. Baehr-Stephan). (2.5-1) The contact temperature is determined by: (2.5-2)



3 Convective Heat Transfer

The passage of heat from a solid wall to a moving fluid (or vice versa) is referred to as heat transfer.

Since the choice of the coordinate reference system is arbitrary, all statements apply analogously to a body moving in a fluid at rest.

3.1 Heat Transfer and Influencing Parameters According to Newton1, the following definition equation applies to the passage of heat by convection:

WFl TTAQ or WFl TTq (3.1-1) where

A

Qq

(3.1-2)

Since, for flowing media, the no-slip condition2 applies to fluid particles on the wall, heat is conveyed in a thin "underlayer by molecular heat conduction.

Idealy With a thermal boundary layer Figure 3.1: Model Presentation of the Temperature Profile in Fluid by Convection (Figure: Wagner)

Accordingly, the heat transfer can, in principle, also be described by the formulas for the conduction of heat, the problem being the unknown thickness of this fluid layer on the wall. In the proximity of the layer on the wall (referred to as boundary layer), the temperature of the fluid changes and assumes a linear course in this boundary layer. Accordingly, analogously to

the boundary layer of the flow ( ) a thermal boundary layer can be established and

therefore:

Fl WQ A T T

(3.1-3)

1 Sir Isaac Newton (1642-1727) recognized this relationship in 1701

2 With the exception of the field of the Knudsen- and the molecular flow. The following statements do not apply there.



The following is a more detailed view of the transfer of heat by convection: Therefore, in principle, the heat transfer coefficient can be calculated knowing the fluid property and the thickness of the thermal boundary layer . By comparing (3.1-3) with

(3.1-1) it follows directly that:

or

(3.1-4)

In Chapter 1, the heat transfer resistance is given by: (3.1-5)

In a few cases (laminar flow in simple bodies) the thickness of the boundary layer can be

calculated accurately analytically. In this case, the values for heat transfer are also established analytically. In the most technically important cases, only a theoretical functional relationship can be identified and specified for delineation by equations. The accurate assimilation of the calculation formula is by comparison with measurements and appropriate correlation formulas.

3.1.1 Parameters Influencing the Transfer of Heat The values for heat transfer vary greatly, depending on the medium and the flow.

Wall T



As first typical values for the heat transfer, the following values can be assumed for approximate values of the order of magnitude:

Heat Transfer Values in W / m K

Typical Attainable

Gases and Vapors Free convection 8 15 5 25 Forced convection 20 60 12 120 Water

Free convection 200 400 70 700 Forced convection 2 000 4 000 600 12 000 Evaporation ca. 4 000 2 000 12 000 Film condensation ca. 6 000 4 000 12 000 Dropwise Condensation - 35 000 45 000 Viscous Liquids

Forced convection 300 400 60 600 (according to Cerbe/Hoffmann and others)

The most important parameters that influence convective heat transfer are:

Geometric shape (plate, sphere, etc.)

Material data ( , , pc )

Thickness of the boundary layer influenced by Laminar or turbulent flow Free or forced flow

Wall roughness

Start-up conditions (hydraulic and thermal)

Strength and direction of the heat flow

Temperature-dependent material values Gases and liquids With or without phase change

Phase change with an increase or decrease in volume

Phase change with or without gravity

r



3.1.2 Dimensionless Characteristic Values

Not used for steady state heat conduction because the number of parameters is small (only ) direct formula

For unsteady state Heat Conduction (HC), additionally t, , cp more complex Parameters reduced to ( , )f Fo Bi Loss of expressiveness; but formulas manageable

Even more complex for Convective Heat Transfer (CHT), therefore no more dimensionless temperature, but dimensionless HT value Nu = f(Re, )

Nuelt-Number1

Fluid

L HTNu

HC

(3.1-6)

HT coefficient KmW Thermal conductivity KmW of the fluid L Characteristic length [m]

Depending on the body (diameter, running length, length through which there is flow,) Once the Nu is determined (by correlation), is also immediately known Note: For the Nu, as well as for the Bi, always observe the exact definition of the correlation

formula! Nu compares the HT with the HC. Always > 1 (real 5 50.000) Reynolds Number

inertial forces

Re friction forces

u L

( = kinematic viscosity) (3.1-7)

Prandtl Number

portheat trans

transportpulsePr

pc

a (3.1-8)

Grashof Number

forces inertial

forcesbuoyancy thermal

gLTGr (3.1-9)

= Volume expansion coefficient K1

for ideal gases: 1

T

Peclet Number

Energy conveyance by flow

Energy conveyance by heat conduction

u LPe

a (3.1-10)

Pr Re

1 Wilhelm Nuelt, 1882 - 1957, german physicist, TH Karlsruhe, TH Mnchen



Stanton Number

heat flow

heat capacity flowp

Stu c

(3.1-11a)

Flwpw

TTcu

qSt

(3.1-11b)

PrRe

Nu

St

This can be converted to (refer to 3.2.1.1)

uSt w

(3.1-11c) Rayleigh Number

Prconductionby nsport energy tra

buoyancyby nsport energy tra

Gr

cgLTRa

p

(3.1-12)

Further characteristic numbers WuSt - F1a



3.2 Heat Transfer with Forced Convection Depending on the application, there is particular interest for

totalQ (or for T ) average values for heat transfer number adequate / meaningful

localT (or localQ ) local or locally averaged values necessary

3.2.1 Correlations for Average Values Refer to HB-K-01 with the equations 3.2-1 to 3.2-6 Some of the formulas can be derived analytically exactly from the Boundary Layer equations (BL equations) (refer to references, such as Merker/Eiglmeier). For example, the so-called Reynolds analogy can be derived from a comparison of the flow BL with the thermal BL (refer to Chapter 3.2.2.1). With that, or by comparison of the definitions, it follows that:

St Re Pr (3.2-7)

Where 1Pr (that is, when the thermal and the hydrodynamic boundary layers have the same thickness) the local heat transfer coefficient x can be derived from the local coefficient of

friction fc and then, by integration over the running length, the average value m can be

calculated. For Pr = 1, L

mm

LNu Re664,0

then results, refer to equation 3.2-4 in HB-K-01.

Apart from the equations from HB-K-01, which are extensively generally valid, there is a large number of special formulas in the literature for more limited applications. There are, for example, the two most frequent special cases: Heating with condensing steam

Heating with a constant heat flow Wq

According to Petukhov (1970), the following empirically improved correlation equation is valid for heat transfer in channels, through which there is turbulent flow at a constant heat flow

density q

187,1207,1

83/2,

,

PrSt

PrRe

Nuqm

qm

(3.2-8a)

with the limits

2000Pr5,010Re1064

According to Merker, for gases in the range 9,0Pr6,0 , this equation can also be replaced

by

29,0PrRe012,05 83,0, qmNu (3.2-8b)



For the case with a constant wall temperature T, according to Gnielinski (1984) the applicable correlation is

3/2

3/2,1

187,121

10008

L

d

Pr

PrReNu Tm

(3.2-9)

with the limits

1010Pr5,010Re2300 46 L

d

For gases, where 7,0Pr and 510Re2300 , Merker specifies a simplified relationship:

8,05,0

, RePr021,0 TmNu (3.2-9)

The pressure loss coefficient is to be inserted in most equations. For either the so-called

Prandtl-Nikuradse equation 3.2-10a (3.2-10a) or the simplified equation 3.2-10b of somewhat limited validity is used.

264,1log82,1 Re , 64 105Re10 (3.2-10b) The magnitude of the Nuelt number is evident from Figure 3.2.

Figure 3.2: Average Nuelt Number for Flow in a Pipe

with wq = constant (Figure: Merker/Eiglmeier)

bergangsbereich = transition range vollturbulent = fully turbulent



3.2.2 Analogy Between Heat and Pulse Transport Why are Analogies used? Friction results from the exchange of pulses between molecules Heat conduction is the transfer of energy between molecules Flow intensifies the effect of heat conduction by convection Therefore, it is obvious that the effects of friction and heat transfer are linked. In laminar cases, simple flows can be described accurately analytically and, therefore, this can also be expected for the transfer of heat. In the case of turbulent flow, only approximate solutions can be achieved in accordance with the non-accurate solutions of turbulent flow. However, these provide the theoretical basis for suitable correlation equations, with which the majority of parameters and measurements can be "compressed" to relevant quantities. Difference Between Flowing Around and Flowing Through In the case of bodies, around or over which there is flow, the flow field is divided1 into an

- outer area of undisturbed flow, with no friction and an - area in the proximity of the wall, in which there is considerable friction. There is

adhesion to the wall In the case of channels and devices, through which there is flow, the flow can likewise frequently be divided into

- a boundary layer flow in the proximity of the wall and - a main flow, which is determined by the geometry (e.g., dividers or nozzle/diffuser) and

also by the boundary layer itself (e.g., with or without detachment in the diffuser).

In the case of pipes2, after an initial distance, the velocity profile can be considered as two boundary layers, which contact each other in the center. Therefore, with appropriate adjustments, the results, determined at a plate boundary layer (body with a flow over it) can be transferred to flow through pipes (body with a flow through it).

Figure 3.3: Intake with Laminar Pipe Flow Figure : Merker/EiEqnmeier)

1 This division is not permissible for very small Reynolds numbers, because here the effect of friction extends far into

the flow that is not adjacent to the wall. 2 And for other channels with a uniform cross section.

hydrodyn. Einlauflnge = hydrodynamic intake length Hydrodyn. ausgebildet = hydrodynamically formed Thermische Einlauflnge = thermal intake length Thermisch ausgebildet = thermally formed



For a detailed description of flow, differentiation is made in Fluid Mechanics between the following boundary layers:

- Boundary layer thickness 0 (for: 99,0/)( 0 uu )

- Displacement thickness - Pulse loss thickness - Energy loss thickness

- Thickness of the viscous underlayer * with a turbulent boundary layer

Due to the heat flow, a temperature profile, which is described similarly to the thickness of the

flow boundary layer 0 , is formed by the

- Temperature boundary layer thickness

Figure 3.4: Flow and Temperature Boundary Layer with the Plate (Pr < 1)

Corresponding to the definition of the Prandtl Number (Eq... 3.1-8) as the ratio of the kinematic

viscosity to the temperature conductivity, it is obvious that, when 1Pr a , the flow

distribution and temperature distribution in the boundary layer are similar. At low Prandtl Numbers, the flow boundary layer is thinner than the thermal boundary layer; at large Prandtl Numbers, the flow boundary layer extends far beyond the thermal boundary layer. The different fluids are sometimes divided approximately into the following classes. The Prandtl Number with the actual material values must always be used for calculations.

1Pr Liquid metals (Pr = 0,005 to 0,05)1 1Pr Gases (air: Pr 0,7 ) 10Pr Liquids (water 2 at 8C) 100Pr High viscosity liquids (up to Pr > 104 to 105 for thermal oils at 0C)

Figure 3.5: Spectrum of Prandtl numbers (Quelle: Jischa)

1 e.g. sodium: Pr 0,007, mercury: Pr 0,02

2 at the melting point Pr 13, at 20C Pr 7, at the boiling point Pr 1,8 (liquid) or. Pr 1,0 (vapor)

Flssige Metalle = Liquid metals Gase = Gases Wasser` = Water le = Oils Organische Flssigkeiten = Organic liquids Quecksilber = Mercury Luft = Air Motorl = Engine oil



When there is laminar flow in a pipe, the velocity distribution is known from Fluid Mechanics (incompressible):

22R r dp

v r 1 (1)4 R dx

(3.2-11)

where the volumetric average

2

m

R dpv (2)

8 dx

and the resulting velocity distribution over the cross section

2

m

v r r2 1 (3)

v R

After

inserting in the energy equilibrium

integrating

adapting to the boundary conditions it follows with the caloric average temperature Tm

2

0

12 v( ) ( )

v

R

m

m

T r r T r drR

(4)

after prolonged calculation (e.g. Merker-Eiglmeier, Bosnjakovic II)

2 v11

48

mm w

RdTT T

dx a

(5)

and where the definition of the number after some transformations

DTT

q

mw

w

11

48...

(6)

48

4,3636 4,3611

L DNu

(7)

that is, when laminar flow has developed in the pipe: Nu = constant

Previous definition: Fl

LNu

where WFlq (8)

compared with thermal conduction: Fl Wq

(9)

(3.2-12) That is, the Nuelt Number can also be interpreted as the ratio of the characteristic length to the thickness of the thermal boundary layer.

Symmetry in the center of the pipe

drdTq

Rrw



The above derivation is valid for / .dT dx const , that is, for a constant heat flow ( wq const )

When Tw = constant, Tm increases exponentially and asymptotically approaches the wall tempe-

rature Tw 0w mT T and 0W

dT

dr (but, the quotient has a fixed limit value)

at the wall As mentioned at the beginning of the chapter for laminar flow, a thermally developed flow is to be anticipated in the case of hydrodynamically developed flow, because of the strict analogy between pulse exchange and heat exchange. This becomes evident in the constancy of the Nuelt Number, which, in the laminar case, depends only on the thermal boundary conditions and not on the Reynolds Number.

Where there is turbulent flow in pipes, the following applies for the shear stress and for the heat flow density:

1v

d

dy (3.2-14a)

1

a

a

dTq

dy (3.2-14b)

The turbulent viscosity and the turbulent thermal diffusivity a depicts the increase in the

exchange due to the turbulent fluctuating movement.

To derive an analogy, the boundary1 layer may now be divided into a

fully turbulent boundary layer and aa and into a

viscous underlayer and aa

1 The flow region primarily is split into the main flow (outer flow) and the boundary layer.

, 4,3636qNu (3.2-13a)

developed flow

.constq

, 3,6568TNu (3.2-13b)

.WT const

T

x

TW

Tm

=const

T

x

TW

Tm

TW=const



3.2.2.1 Reynolds Analogy (Pipe Flow) For the Reynolds analogy, the viscous underlayer is neglected. Within the turbulent boundary layer y*, the velocity increase is assumed to be linear, from zero at the wall up to the

volumetric average velocity mv .

In the fully turbulent boundary layer, equations. 3.2-14a and -14b reduce to:

vd

dy (3.2-15)

pdT

q c ady

(3.2-16)

and, with the assumption: Pr 1 or a *0 y (3.2-17)

and by dividing the formulas 1 v

p

d

q c dT

(3.2-17b

With the linear increase of v and T in the boundary layer from y = 0 to y = y*

(3.2-18)

With the generally valid equation (refer to Fluid Mechanics)

2

8w v

(3.2-19)

the St number (3.1-11b) is as follows

2v v v v

w p w mw w

m p w m m m p w m m

c T TqSt

c T T c T T

or. 8

St

(3.2-20)

Reynolds

Because of the assumptions, 3.2-20 is valid only for:

Pr 1

Re ( const in the Nikuradse/Moody diagram)

average value of turbulent fluctuations (not the average value over the cross section, vm).



3.2.2.2 Prandtl Analogy (Pipe Flow) Improvement resulting from the consideration of the viscous underlayer

For the viscous underlayer, it follows from Eqn. 3.2-14a and 3.2-14b, and a being neglec-

ted, that:

v Pr v

w

w p p

d d

q c a dT c dT (3.2-21a)

Integrating from y = 0 to y = y1

1

1

Prw

w p w

v

q c T T

(3.2-21b)

Furthermore, assuming that: = a ( Prt = 1) in the fully turbulent boundary layer, Eqn. 3.2-18 applies analogously

m 1

1

v -v1

w

w p mq c T T (3.2-22)

Eliminating T1 and inserting in the definition of the St Number (3.1-11b) gives: (3.2-23)

For Pr = 1 both analogies are identical!

From the Universal Wall Law (refer to Fluid Mechanics) the boundary of the viscous underlayer can be derived with Eqn. 3.2-19:

1

m

v10,8

v 8

(3.2-24)

and, with that, it follows that:

8

1 10,8 Pr 18

St

(3.2-25)

Prandtl

Because of the assumption that Prt = 1 valid only for large Re numbers Because of the two-layer model only for medium Pr numbers (Pr 0,5 5) Nevertheless, physically correct basis for more accurate correlation equations (such as 3.2-8a)

y

y1

y*

r

Main flow



3.2.2.3 Von-Krmn Analogy (Pipe Flow) Improvement by introducing a transition range. According to Jischa it follows that

8

5 Pr 11 5 Pr 1 ln

8 6

St

(3.2-26)

von Krmn

This three-layer model is now applicable to the whole range of the Prandtl Numbers (theoretically).

3.2.2.4 Analogies in the Plate Boundary Layer With appropriate transformation, the formulas for heat conveyance in a pipe can be used as an approximation for the plate boundary layer. By comparison, the following can be derived from the basic formulas of fluid mechanics:

4 wc or. 8 2

wc (3.2-27)

after insertion in 3.2-20 (8

St ) it follows that

2

wc

St (3.2-28)

Reynolds

or in the Prandtl analogy (3.2-25) adapted to the values measured:

2

1 13,2 2 Pr-1

w

w

cSt

c (3.2-29)

or according to Petukhov and Popov (1963):

23

2

1 12,8 2 Pr -1

w

w

cSt

c

(3.2-30)

transition area,

and a a



Friction coefficients Different formulas, some of which can be derived analytically, some partly analytically and partly improved by adapting them to measured values, can be used for the coefficient cW or (in the case of local consideration) for cf(x). At the same time, it is necessary to differentiate between: A local or an averaged1 coefficient:

The coefficient of friction, cf or cW, is defined as ( u = velocity of the undisturbed outer flow)

2

2

)()(

u

xxc Wf

or:

2

2

u

c WW

(3.2-31)

Laminar or turbulent boundary layer:

The Reynolds Number is formed with the running length x. A laminar boundary layer exists

up to approximately 5105Re x to

610 , xux Re

In the absence of further information, a transition point with 610Re u is to be assumed for

exercises carried out within the scope of this course of lectures. Laminar Boundary Layer (plate) According to Blasius (1908), the ("exact") solution of the boundary layer equations is given by:

5,0Re664,0)(

xf xc (3.2-32a)

5,0

Re328,1

LWc (3.2-33a)

Turbulent Boundary Layer (plate, hydraulically smooth surface) 610Re x

The configuration and factors of the formulas vary according to the author, compare HB-F04. Unless stated otherwise, the Jischa formulas are to be used for exercises and examinations:

Jischa 2,0

Re0577,0)(

xf xc (3.2-34)

2,0

Re072,0

LWc (3.2-35)

1 The mean value is calculated by integrating the friction forces along the running length of the plate.

( )R R W W totF dF x dA A , dxbdA und totA b L , b = Plate width



3.2.3 Local Heat Transfer Values at the Plate On the one hand, the Stanton Number, which was introduced in the previous chapter, in conjunction with the local coefficient of friction cf(x), is adequate for determining local heat

transfer values x over the running length of a flat plate. However, since the Reynolds Number

is already generally known (e.g., for clarification of whether the flow is laminar or turbulent), it is also possible to compile a correlation formula Nu = f(Rex) here for the flow problems given.

3.2.3.1 Laminar Boundary Layer From the differential equations for the plate boundary layer, the following can be derived for the local quantities at location x (compare also Prandtl analogy):

)( rPfeR

Nu

x

x

where /xNu and /xueR , x = coordinates in the direction of flow

Where rPeRStNu it follows for Pr=1 with Eqn. 3.2-28 and 3.2-32

xxxxxxeReRStrPeRStNu 332,0

or:

Pr=1 332,0

x

x

eR

Nu

(3.2-36)

For Pr 1, the following formulas can be applied:

)10Pr6,0( 3/1Pr332,0

x

x

eR

Nu

(3.2-37)

According to Specht and Jeschar (1984)

)Pr0( 25,03/23/1

Pr12,092,0

Pr332,0

x

x

eR

Nu

(3.2-38)

According to Kays und Crawford (1980), the following applies at constant heat flow in the range of the Prandtl analogy for the laminar boundary layer

Wq constant Pr 1 3/1Pr453,0

x

x

eR

Nu

(3.2-39)



3.2.3.2 Turbulent Boundary Layer Since turbulent heat exchange in the boundary layer is significantly higher than heat

conduction in the fluid, heat transfer for constqW is only a few percent higher than for

constTW . A differentiation, according to the nature of the boundary condition, is

therefore not necessary and the formulas of the analogy can be applied with the appropriate cf values. According to Merker/Eiglmeier, the following easier equation can be applied for gases in the

range 1Pr5,0 .

0,8 0,6Nu 0,0287 Re Prx x (3.2-40a)

The average value over the plate (comparable with, or as a replacement for, Eqn.3.2-5, HB-THFL-K1) is obtained by integrating over the running length

0,8 0,60,0357 Re Prm LNu (3.2-40b)

The transition point from a laminar to a turbulent boundary layer depends, for instance, on the boundary conditions (shape of the front edge of the plate, turbulence of the incident flow, etc.) and can therefore not be determined as accurately as it can in the case of steady state pipe

flow. Assuming that the transition point starts at 5102Re x , an approximation equation,

for which it is no longer assumed that the boundary layer is turbulent from the start of the plate, can be composed by appropriate regional integration and addition of the formulas for the local laminar and the local turbulent boundary layer. According to Merker/Eiglmeier, the following equation applies to this:

9400RePr036,0 8,06,0 LmNu (3.2-41)

3.2.4 Heat Transfer at the Cylinder with Transverse Incident Flow (not yet in this term)

3.2.5 Heat Transfer at the Pipe Inlet (not yet in this term)



3.3 Heat Transfer with Free Convection 3.3.1 Free Convection

Simple formulation according to Schlnder: potential energy of the buoyancy forces: a) kinetic energy of the buoyancy flow: b) K = correction factor for friction by wall adhesion

Comparison with measurements 5K

Figure 3.3: Temperature and Velocity Profile for Free Convection (here: Pr < 1) The equilibrium conditions for steady state flow requires (3.3-1)

3

w

w

g L

Gr (3.3-2)

At constant pressure: T

(3.3-3a)

and, for ideal gases, TRp

because Wp p

T

1 , volume expansion coefficient (3.3-3b)

w(y)



For a flat plate we already know a heat transfer formula (with forced flow, HB-K01, Eqn. 3.2-4):

3 PrRe664,0 Nu

and with 5,2

ReGr

(Eqn. 3.3-1) it follows theoretically that

34 Pr528,0 GrNu

According to Mayinger, when adapted to measured values (and theoretical formulation) the following applies:

4

4

0,55 Pr

0,55

Nu Gr

Ra (3.3-4)

laminar boundary layer

valid for 0,5 < Pr < 1000

4 910 < Gr Pr < 10

valid for Plate of height L Horizontal cylinder, if the Gr number is formed as follows: L3 L L2

cylinder diameter length over which there is flow 2'L D

According to Wagner, for a turbulent boundary layer:

30,14 PrNu Gr (3.3-5)

turbulent

valid for 9Gr Pr > 10

Reference temperature for properties

refT T for (volume expansion)

2

wref

TTT

for all other properties

Further geometries/special cases WuSt-K-02, Eqn. 3.3-6 to 3.3-14



3.3.2 Superimposition of Free and Forced Convection

w

0w bouyancyw w bouyancyw w

Two approaches

a) 2 2 2Re Re Re Re2,5

free forced forced

Gr (3.3-17)

0,5 < Pr < 2500 and 0,1 < Re < 107

Not applicable for - flow in opposite directions or - Refree Reforced unstable states!

b) 3 33 free forcedNu Nu Nu (3.3-18)

(-) for flow in opposite directions Nu dependant on the geometry, etc.

3.4 Heat Transfer During Phase Change For heat transfer with a phase change, the heat conveyance is linked to the conveyance of the material. Heat transfer with a phase change is important for:

3.4.1 Heat Transfer During Evaporation There are many distinct possibilities for heat transfer during evaporation. The average velocity of the main flow during the phase change is one significant differentiation characteristic.

Container Boiling Flow Boiling



3.4.1.1 Container Boiling Evaporation of fluids at rest TW TS low TW TS larger TW TS very high



Principle Relationships: a) Convection boiling: such as free convection

laminar: 114c T (3.4-1)

turbulent: 213c T (3.4-2)

b) Nucleate boiling: 33c T (3.4-3)

c) Film boiling: various (complex!)

To a) Convection Boiling

- Laminar boundary layer: 41

Pr)(60,0 GrNu (3.4-4)

- Turbulent boundary layer: 31

Pr)(15,0 GrNu (3.4-5)

Valid if 2 < Pr < 100

Laminar changes to turbulent at approx. 107 < Gr Pr < 108 (intersection at 2 107) Characteristic length: - Round plate L = D

- Horizontal pipe L = D - Other geometries L = length, over which there is flow Dispersion range approx. 20%

To b) Nucleate Boiling

Buoyancy is determined by the number and size of the bubbles Affected by numerous parameters, including. - The roughness of the heating surface - The material of the heating surface ( contact angle) - Pressure - Material data According to Stephan, when p = 1 bar:

7 / 3

0

S

Tc

T (3.4-6)

where 0 ( ; ; ; ; ; ; ; )f f D v s pc f h T R

Surface tension Wetting angle Roughness When p = 1 bar and RP = 1 m (finely sized)

Water: Km

Wc

6

0 1061,10 KCTs 1,373100

Ammonia (NH3): KmWc

6

0 1051,8 KTs 7,239



To c) Film Boiling

A vapor film is a poor conductor T very high under some circumstances thermal radiation is relevant.

numbers for container boiling HB-K-03

3.4.1.2 Flow Boiling Evaporation with forced flow (mainly in a pipe) - more complex than container boiling - further phenomenon: subcooled boiling (at Tliquid < Tboiling)

number without radiation



3.4.2 Heat transfer with Condensation The reverse of the evaporation process, condensation (liquefaction of a saturated vapor) is also a form of two-phase flow with linked conveyance of heat and material. Because of the large reduction in volume during condensation, buoyancy is not the most relevant problem. The fundamental problem is the behavior of the liquid phase at the wall, under the influence of gravity, which may, however, be influenced analogously to the evaporation process, by the sheering stresses of the gas phase, which act at the phase interface. Condensation can be divided into

a) Diffusion of the vapor to the phase interface

b) Exothermic phase change (heat of condensation is released)

c) Heat conveyance to the cooled wall A differentiation is made between:

A) Film condensation B) Droplet condensation

The nature of the condensation depends on the wetting relationships of the wall. For example, a coherent condensation film can only be formed on a surface that is smooth, fat-free and clean. Mainly there is mixed condensation. The presence of gases, that do not condense, impedes the conveyance of the vapor to the phase interface and, with it, the transfer of material and heat. Once again, the influence of the geometry is important, that means, whether the flow of vapor in the pipe is horizontal or vertical, or around the pipe or pipe nest. Similarly, the state of flow (laminar or turbulent) in the film and in the gas phase affects the conveyance of the material and pulse. Furthermore, contamination of the surface of the condenser also has an influence. This problem is very complex, because the contamination (with the thickness sc and the thermal

conductivity value c), as a resistance to the conduction of heat, not only impedes conveyance of the heat, but also deteriorates wettability significantly and, therefore, affects the film or dropwise condensation. Because of the large number of configurations and operating parameters, reference is made here only to the literature (e.g. Chapter J in the VDI-Wrmeatlas). Some basic equations are given in HB-K-03.

3.4.3 The Heat Pipe (explain only verbally in this term)



Derivation of universal wall law, eq. 3.2-24 measurements for laminar and turbulent flows give for

laminar flow: u y

turbulent flow: 1

u ln y Ck

with k 0,41 C 5,0

with definitions:

u

uu

and W u

it follows Wu yy

y

u Wandschubspannungsgeschwindigkeit = velocity correlated to wall shear stress

point of intersection at u y 10,8 (if k 0,41 C 5,0 )

or u y 11,6 (if k 0,40 C 5,5 )

limit/edge of viscous underlayer at

u

u y 10,8u

resp. u 10,8 u

with definition of Wu

and with eq. 3.2-19: 2W

mv8

W mu 10,8 u 10,8 10,8 v8

resp. m

u10,8 eq. 3.2 24

v 8


Schelling Copying permission only for internal use 4-HTR-Chap04-V08-2015.03.07.doc

4 Heat Transmission

The concepts of heat transmission, TAkQ and heat resistance, R T Q , have already been repeatedly mentioned in previous chapters. The two concepts comprise of, or describe the collective action of several effects, for example

A typical application is the transfer of heat between two fluids of different temperatures, which are separated by a solid wall. Very frequently, the individual wall temperatures are not of concern, but only the heat flowing between the fluids at a given temperature difference or, conversely, the essential temperature difference for a desired heat flow. Example:

Pipes in a heat exchanger: House wall:

In the steady state case, according to the first law of thermodynamics, the same amount of heat flows through all layers. The transfer of heat, internally and externally and the thermal conduction, in one or more layers, is depicted by a single coefficient

Q k A T (4.1)

T

RQ

or

TQ

R (4.2)

where

k Heat transmission coefficient W m K

A Reference surface m

T Temperature difference of fluids K

R Heat resistance K W



Since the reference surface can sometimes be defined differently1, there is no point in stating

the k value without defining the surface. For this reason, frequently it is also simply given as

the product Ak . From equations 4.1 and 4.2, it is evident that:

1

Rk A

or 1

k AR

(4.3)

The location with the highest thermal conductivity resistance or heat transfer resistance is decisive for the heat flow (or the temperature difference that results from it). Example: Heat flow in a pipe Comparison with mass flow in a car radiator

4.1 Heat Transmission Laws Compare Chapter 2.2.3 (1D heat conduction with heat transfer) Addition of all resistances for

- Heat transfer A

R

1

- Heat conduction 1

s

sR

AA

,1 1

1n m jtot

i ji i j j

sR

A A

(4.4)

1The external diameter of a pipe can, on the one hand, be measured easily, however, frequently the

internal diameter is determined by the mass to be conveyed and the external diameter ensues from the thickness of the insulation. A flat surface may refer to the active surface, or to the total surface, with frame and holding device, or front and back, etc.

Water Air

T



For this reason, or with Eqn. 4.3, it follows for a

a) Flat wall

Ai = Aj = A

1 1

1 1

1n m j

i ji j

ksR A

(4.5)

Example: Old double window: 1, 1, 2, 3, 2, 4

normal thermal pane: 1, 1, 2, 3, 2

superinsulation: 1, 1(sheet), 2(air), 1, 2, 1, 1, 2

For a flat wall, the convective heat transfer can also be specified by the thickness of a so-called substitute wall:

Where TAkQ

and AA

s

AAk ai

111

According to the sketch, it follows that the thickness of substitute wall is:

i a

sk

(4.6)

Heat transmission can now be calculated again with a simple Fourier equation.

b) Curved Walls

Area Ai Aa Reference area is arbitrary, in principle any

Frequently related to the external surface (because the highest resistance is frequently there)

With reference to the exterior (subscript a)

m

a

i

a

ia

a

A

As

A

Ak

11

1 Am = Average area (4.7a)

With the average area Am (compare Chapter 2.2.3)

ln

a im

a

i

A AA

A

A

m a iA A A (4.7b+c)

cylinder spherical wall



Insulation has two effects on curved walls:

Increases the thermal resistance

Increases the surface area for convective transfer of heat By neglecting the convective resistance on the interior of a pipe, through which there is flow

(rapidly flowing liquid), then (Eq: 4.1 and 4.7 with i ):

aai

a

ai

rr

r

TTLQ

ln

2

With ra = ri + s and differentiating with respect to ra, the derivative becomes positive at:

1

aa r

An insulating layer then reduces the heat flow only if:

Cylinder: aa r (4.8a)

Analogously for a sphere: 12 a a

r (4.8b)

For cylinders, can be transformed dimensionless:

*

0

1

2

1ln 1

1

i a

i a i

i

QQ

L T T

s

r r s

r

Parameter B

Where *

0Q = Heat flow with no insulation is given in Figure 4.1:

a i

rB

Figure 4.1: Effect of Insulation on the Heat Flow for a Pipe (Scource: Merker/Eiglmeier)

4.2 Intermediate Temperatures Temperatures of the individual layers of the wall can be calculated with the appropriate

formulas for heat transfer and heat conduction, commencing from one side

Graphic method (see exercise 2.6)

i

s

r



5 Heat Exchanger In the introductory Chapter 1, heat exchangers were divided into two1 main groups: Regenerators Recuperators For all heat exchangers, the heat flow is principally determined by the highest resistance (refer to Chapter 4). For air-water heat exchangers, this is mainly the air side, that is, this side is decisive for the dimensioning of the heat exchanger.

The formulas previously described for calculating the heat transfer generally address special cases, in which the driving temperature difference between the wall and the fluid (or between two fluids separated by a wall) was known and was constant over the area of the heat exchanger under consideration. In general, this is not the case for industrial heat exchangers, for which the temperatures of the two mass flows mainly change continuously.

For the simple geometries of parallel and reverse flow devices, the formulas previously developed can nevertheless be used, whereby, only the driving temperature differences have to be appropriately determined.

For more complex geometries, a more expensive method is required to determine the Number of Transfer Units (NTUs)

5.1 Parallel and Reverse Flow Heat exchangers Reverse flow HX Parallel flow HX

1Another type is the mixing heat exchanger, where the mass flows partly or completely mixes e.g., in an evaporation-

cooling tower. Evaporation is addressed in Chapter 8.



Local: localbalocallocal

kq (1)

Integral: dAqQA

local (2)

Statement: mAkQ (3a)

Assumptions: - No change in the fluid properties ( thus, no phase change) - Constant mass flow ( mass flow branching not permitted)

dAkQd m (3b)

from (3a) (2) (1)local local a b local

m

q dA k dAQ

k A k A k A

1

m a b localdA

A (5.1-1)

How does ba change over the running length? First Law of Thermodynamics

Without insulation losses: ba QQ (4)

Without a phase change: pdQ dH m c d

aa a p a

dQ m c d a

aa

a p

dQd

m c

(5)

Analogously for b) b

bb

b p

dQd

m c

(6)

(5) (6)

1 1

a b

a b

a p b p

d d d dQm c m c

(3 ) 1 1

a b

b

a p b p

d k dAm c m c

(7)

For the total heat exchanger the first Law of Thermodynamics is valid

1221 bbpbbaapaa ba cmQcmQQ (8)

apa

aa

cmQ

121 and

2 1 1

b

b b

b pm cQ

(9)

in (7) in (7) Combining (7) and (9), it follows that:

1 2 2 1

a a b bd k dAQ Q

(10)

and by division by , re-sorting and bracketing follows

1 2 2 1

a b a b

d kdA

Q (11)



ln

big

big small

small

kA

Q (11)

respectively:

Q (12) Compare with statement (3a): average logarithmic temperature difference

ln

big small

m

big

small

(5.1-2)

mAkQ (5.1-3)

For calculating k see chapter 4, Heat Transfer. For parallel flow Heat Exchangers, an analogous calculation gives the same result. For cross flow heat exchangers

- m and a correction factor are used for calculation.

- the larger the number of multiple passages, the smaller the correction.

5.2 Number of Transfer Units (NTU) Dimensioning a heat exchanger is determined by the following:

If the velocities and external heat losses are neglected, the First Law of Thermodynamics, for a steady state, reads:

1 1, 1, 2 2, 2,0 m h - h + m h - h I O I O

Kreuzgleichstrom-WT = Cross-parallel flow heat exchanger Kreuzgegenstrom-WT = Cross-reverse flow heat exchanger



where pc const it follows that:

1,I 1,O

2,O 2,I

T T

T T

(5.2-1)

pW m c = Heat capacity flow, T positive (5.2-2)

With the kinetic statement TAkQ it follows that

2211 TWTWAkQ m (5.2-3)

multiple independent dimensionless parameters can be established

(1) (5.2-4a)

or 22

11

1WR

RW (5.2-4b)

(2) 1

thermal task set= requirements from process

O I

m

(5.2-5)

22

NTUW

Ak

is already contained in (1) and (2)

111

11

1

1

2

2

1RNTU

W

RWNTU

W

RAk

WAkNTU i

and (5.2-6)

Dimensionless Average Temperature Difference

(3) m (Index I: Inlet) (5.2-7)

Inlet of w

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HTR LectureNotes 2015 Summerterm