Ahsan A. (ed.) Evaporation, Condensation and Heat transfer

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Heat Transfer in the Transitional Flow Regime

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Numerical Modeling of

Cross-Flow Tube Heat Exchangers

with Complex Flow Arrangements

Dawid Taler

, Marcin Trojan

and Jan Taler

Cracow University of Science and Technology (AGH),

Department of Energy Systems and Environment Protection,

Faculty of Mechanical Engineering and Robotics,

Cracow University of Technology,

Chair of Power Plant Machinery,

Faculty of Mechanical Engineering,

Poland

1. Introduction

Cross-flow tube heat exchangers find many practical applications. An example of such an

exchanger is a steam superheater, where steam flows inside the tubes while heating flue gas

flows across the tube bundles. The mathematical derivation of an expression for the mean

temperature difference becomes quite complex for multi-pass cross-flow heat exchangers

with many tube rows and complex flow arrangement (Hewitt et al., 1994; Kröger, 2004;

Rayaprolu, 2009; Stultz et al., 1992 ; Taler, 2009a). When calculating the heat transfer rate,

the usual procedure is to modify the simple counter-flow LMTD (Logarithmic Mean

Temperature Difference) method by a correction F

determined for a particular

arrangement. The heat flow rate Q



transferred from the hot to cool fluid is the product of

the overall heat transfer coefficient U

, heat transfer area A, correction factor F

and

logarithmic mean temperature difference ΔT

. The heat transfer equation then takes the

form:

ATlm

QUAF T=Δ



(1)

However, to calculate steam, flue gas and wall temperature distributions, a numerical

model of the superheater is indispensable. Superheaters are tube bundles that attain the

highest temperatures in a boiler and consequently require the greatest care in the design

and operation. The complex superheater tube arrangements permit an economic trade-off

between material unit costs and surface area required to obtain the prescribed steam

outlet temperature. Very often, various alloy steels are used for each pass in modern

boilers.

Evaporation, Condensation and Heat Transfer

262

High temperature heat exchangers like steam superheaters are difficult to model since the

tubes receive energy from the flue gas by two heat transfer modes: convection and

radiation. The division of superheaters into two types: convection and radiant

superheaters is based on the mode of heat transfer that is predominant. In convection

superheaters, the portion of heat transfer by radiation from the flue gas is small. A radiant

superheater absorbs heat primarily by thermal radiation from the flue gas with little

convective heat flow rate. The share of convection in the total heat exchange of platen

superheaters located directly over the combustion chamber amounts only to 10 to 15%. In

convective superheaters, the share of radiation heat exchange is lower, but cannot be

neglected.

Correct determination of the heat flux absorbed through the boiler heating surfaces is very

difficult. This results, on the one hand, from the complexity of heat transfer by radiation of

flue gas with a high content of solid ash particles, and on the other hand, from fouling of

heating surfaces by slag and ash (Taler et al., 2009b). The degree of the slag and ash

deposition is hard to assess, both at the design stage and during the boiler operation. In

consequence, the proper size of superheaters can be adjusted only after boiler’s start-up. In

cases when the temperature of superheated steam at the exit from the superheater stage

under examination is higher than its design value, then the area of the surface of this stage

has to be decreased. However, if the exit temperature of the steam is below the desired

value, then the surface area is increased.

2. Mathematical model of the superheater

To study the impact of superheater fouling on flue gas and steam temperatures, a numerical

model of the entire superheater, has been developed. It was assumed that the outer tube

surfaces are covered with bonded ash deposits with a uniform thickness. The temperature of

the flue gas, tube walls, and steam was determined using the finite volume method (FVM)

(Taler, 2009a). The subsequent stages of the superheater were modeled as either cross-

parallel-flow or cross-counter-flow. As an example, a numerical model of the first stage

convective will be presented in detail (Figs. 1 and 2). The first stage convective superheater

is a pendant twelve-pass heat exchanger.

The superheater is constructed of circular bare tubes and is situated at the back of the

second and third stages of the superheater (Figs. 1 and 2). The tube is made of grade St 20

carbon steel, having a 42 mm outside diameter and 5mm thick wall. The superheated steam

and the combustion products flow at right angles to each other. The convective superheater

considered in the paper can be classified according to flow arrangement as a mixed-cross-

flow heat exchanger (Figs. 2 and 3). Each individual pass consists of two tubes through

which superheated steam flows parallel (Fig. 2).

Based on the energy conservation principle, a mathematical model of steam superheater

with 12 tube rows and complex flow arrangement was developed.

The radiant and convective superheaters are located in boiler passes through which high

temperature flue gas flows. The gas temperature drops from about 1100 ˚C at the exit of the

furnace chamber to about 400 ˚C before entering the economizer. Radiant platen

superheaters are located in areas of highest flue gas temperature and the water heater in the

lowest temperature pass.

Numerical Modeling of Cross-Flow Tube Heat Exchangers with Complex Flow Arrangements

263

Fig. 1. 50 MW coal-fired utility boiler with steam flow rate of 210·10

kg/h: T

, T

, and T

denote flue gas temperatures at the furnace exit, after the superheaters and after the air

heater, respectively.

Evaporation, Condensation and Heat Transfer

264

Fig. 2. Arrangement of boiler heating surfaces.

Numerical Modeling of Cross-Flow Tube Heat Exchangers with Complex Flow Arrangements

265

3. Mathematical model of one row tube heat exchanger

A mathematical model of the cross flow tubular heat exchanger, in which air or flue gas

flows transversally through a row of tubes will be developed. The system of partial

differential equations describing the space and time changes of water or steam T

, tube wall

, and air or flue gas T

temperatures are:

()

wrr

∂∂

+=−−

∂

, (2)

ww w

crk

trr r

⎛⎞

∂∂

∂

⎜⎟

∂∂ ∂

⎝⎠

, (3)

()

wrr

∂∂

+=−−

∂

. (4)

The energy balance Equations (2-4) are subject to boundary and initial conditions. The

boundary conditions are as follows:

()

Txt ft

= , (5)

()

in in

wrr wrr

khTT

⎛⎞

∂

=−

⎜⎟

∂

⎝⎠

, (6)

()

wrr mwrr

khTT

⎛⎞

∂

=−

⎜⎟

∂

⎝⎠

, (7)

()

= . (8)

The initial conditions are:

() ()

101,0

Txt T x

= , (9)

() ()

0,0

,, ,

wtw

Txrt T xr

= , (10)

() ()

202,0

,, ,

tTx

++ ++

= . (11)

The symbols

()

ft and

()

ft denote functions describing the variation of the boundary

temperatures of fluids in time. The symbol T

stands for the mean gas temperature over the

row thickness defined as:

()

TTxytdy

++ +

∫

. (12)

Evaporation, Condensation and Heat Transfer

266

The numbers of heat transfer units

N and

N are given by:



, (13)

where:

in in x

AUL= ,

oox

AUL= .

The time constants

, and

are:

= ,

in o

hA hA

= , (14)

where:

11in x

mAL

212 2

mss L

⎛⎞

=−

⎜⎟

⎝⎠

wmwxw

mU L

= ,

()

mino

UUU=+ .

The transient fluids and wall temperature distributions in one row heat exchanger (Fig. 3)

are then determined by the explicit finite difference method.

3.1 Transient model of one row tube heat exchanger

Transient distributions of fluid and tube wall temperatures were determined using an

explicit finite-difference method. The tube wall was divided into three control volumes

(Fig. 3). The finite difference cell is shown in Fig. 3.

Approximating Eq.(2) using the explicit finite difference method gives:

1, 1, 1

1, 1 1,

1, 1 1, 1 ,

1, 1, 1,

nn n

ii wwi

nn n

ii i

TT T

ττ

⎛⎞

−

ΔΔ

⎜⎟

=− − −

⎜⎟

⎝⎠

, 1,...,iN= , 0,1,...n = . (15)

The finite volume method (Taler, 2009a) was used to solve Eq.(3). The ordinary differential

equations for wall temperatures at nodes were solved using the explicit Euler method to

obtain:

()

()()

()

,, ,

1, 1, 1

,1 , , ,

nn n

ww i ww i wm i

nn nn n

wm i wm i ww i ww i

wm i ww i

TT T

ThrkTT T

=+ ×

⎧ ⎫

⎛⎞

⎡⎤

⎪ ⎪

⎜⎟

−⋅Δ − −

⎢⎥

⎣⎦

⎪ ⎪

⎜⎟

−

⎪ ⎪

⎝⎠

×Δ +

⎨ ⎬

ΔΔ

⎪ ⎪

⎩ ⎭

(16)

()

() ()

,, ,,

,, ,

23 23

nn n n

wz i wm i ww i wm i

nn n

wm i wm i wm i

TT T T

TT T t

rr rr

⎡

⎤

−−

⎢

⎥

=+ Δ ⋅ +

⎢

⎥

ΔΔ

⎣

⎦

(17)

Numerical Modeling of Cross-Flow Tube Heat Exchangers with Complex Flow Arrangements

267

()

()()

() ()

,, ,

'''

2, 2,

,2 , ,,

5 3

nn n

wz i wz i wm i

nn n nn

wm i wm i wz i wz i

wm i wz i

z z

TT T

ThrkT TT

=+ ×

⎧ ⎫

⎡⎤

⎪ ⎪

⎢⎥

⎡⎤

+⋅Δ −−

⎪ ⎪

⎢⎥

⎣⎦

⎪ ⎪

⎢⎥

−

⎪ ⎪

⎣⎦

×Δ +

⎨ ⎬

ΔΔ

⎪ ⎪

⎩ ⎭

(18)

Fig. 3. Control volume for one-row cross-flow tube heat exchanger.

Solving Eq. (4) using the explicit Euler method gives the temperature of the fluid 2 at the

exit of the finite volume (Fig.3):

() () ()()

()()

'''

2, 2,

'' '' ' ''

2, 2, 2, 2, ,

2, 2, 2,

nn nn

ii ii wzi

nn n

ii i

TT TT T

ττ

⎡

⎤

⎢

⎥

ΔΔ

⎡⎤

=+ − + −

⎢

⎥

⎢⎥

⎣⎦

⎢

⎥

⎣

⎦

1,...,iN= , 0,1,...n = . (19)

The initial temperature distributions for the fluid 1 and 2 are given by Equations (9-11),

which can be written as:

Evaporation, Condensation and Heat Transfer

268

,0 ,ww i t ww i

= ,

,0 ,wm i t wm i

= ,

,0 ,wz i t wz i

= , 1,...,iN= , (20)

where the symbols

,ww i

T ,

,wm i

T , and

,wz i

T denote the wall temperatures at the nodes.

The boundary conditions for fluids are given by Equations (5) and (8). The presented finite

difference method is accurate and easy to program. In order to assure the stability of the

calculations, the Courant conditions for both fluids and the stability condition for the

Fourier equation defining transient heat conduction in the tube wall, should be satisfied.

4. Mathematical model of convective superheater

The flow arrangement and division of the first stage convective superheater into finite

volumes is depicted in Fig. 4.

The superheated steam and the combustion products flow at right angles to each other. The

first stage of a convective superheater can be classified according to flow arrangement as a

mixed-cross-flow heat exchanger. The superheater tubes are arranged in-line (Fig. 5b). Each

individual pass consists of two tubes through which superheated steam flows parallel. The

tube’s outer surface is covered with a layer of ash deposits.

In the following, finite volume heat balance equations will be formulated for the steam, the

tube wall, and the flue gas. A steam side energy balance for the ith finite volume gives

(Fig. 5a):

, ,1

,,1

,1, ,1

si si

sps si in s wi sps si

mc T d xh T mc T

⎛⎞

+Δ − =

⎜⎟

⎝⎠



, (21)

Rearranging Eq. (21) gives

()

,,1

,1 , 1,

si si

sps si si ins wi

mc T T A h T

⎛⎞

−=Δ −

⎜⎟

⎝⎠



, (22)

where the mesh tube inner surface is

in in

Adx

Δ= Δ

(23)

The steam average specific heat at constant pressure is given by:

() ( )

,,1

ps si ps si

spsi

cT cT

+ +

≈=

(24)

After rewriting Eq. (22) in the form

,1 , , 1,

si s psi s in si s in w i

spsi s in

TmchAThAT

mc h A

⎡

⎤

⎛⎞

=−Δ+Δ

⎢

⎥

⎜⎟

⎝⎠

⎣

⎦

+Δ



1,...,iN=

, (25)

the Gauss-Seidel method can be applied for an iterative solving nonlinear set of algebraic

equations (25). Introducing the mesh number of transfer units for the steam:

() ( )

,,1

sin sin

spsi

spssi pssi

hA hA

mc T c T

ΔΔ

Δ= =

⎡

⎤

⎢

⎥

⎣

⎦



(26)