Chapter 4 Pressure Drop in Channels and Heat Exchangers

Learning Objectives

At the end of this chapter the student should:

• Recognise the importance of pressure drop in heat transfer system design

• Know the three mechanisms governing pressure drop, i.e. gravity, fluid

acceleration and friction.

• Be able to use correlations to determine the magnitude of the pressure drop in

single phase flows.

• Be aware of the strategy used in determining pressure drop in two-phase flows.

4.1 Introduction

As a fluid flows through a heat exchanger there will normally be a pressure

drop in the direction of the flow (in some special situations where the fluid

velocity decreases there may be an increase in pressure). Pressure drops

occur in the flow channels, nozzles, manifolds and turning regions in the

headers of heat exchangers and each of these pressure drops must be

evaluated, unless experience suggests that one or more may be neglected.

As we have seen in section 3, when deriving Reynolds Analogy, there is a

relationship between heat transfer coefficient and frictional pressure gradient.

The designer will therefore find that measures to increase heat transfer

coefficients tend to also increase the frictional pressure gradient. This does

not necessarily have an adverse effect on overall pressure drop - clever

design may mean that an increased pressure gradient is outweighed by a

decrease in required passage length so the overall pressure drop remains

acceptable.

The designer should aim to “use” all or most of the allowable or available

pressure drop. Having a lower than permissible pressure drop across the

heat exchanger implies that improvements could be made:, depending upon

the application e.g. a longer heat exchanger would have a higher effectiveness

4.1

allowing greater heat recovery; changes in the heat exchanger geometry (for

example, use of smaller tubes) would result in a more compact design for a

particular heat load.

Determination of the maximum allowable pressure drop involves practical

and thermodynamic considerations. If starting from first principles, it is

possible to pursue various strategies to minimise entropy generation within a

heat exchanger, remembering that entropy is generated by irreversible

pressure drops in the heat exchanger and by heat transfer through a finite

temperature difference. From purely thermodynamic considerations,

frictional pressure drops should be minimised since they are irreversible,

while in principle the work done in accelerating and raising the fluid may be

recovered. An entropy analysis is particularly relevant if considering heat

exchangers in power or refrigeration plant. For many heat exchangers the

inlet and outlet temperatures and flow rate are fixed (e.g. an oil cooler) and

the allowable pressure drop is a function of the pumps and fans used. There

may, of course, be a trade off in this situation; for example, it may be decided

to uprate a fan to permit the use of a more compact or cheaper heat

exchanger. In applications involving natural convective circulation the

pressure drop and flow rate are determined by the geometry of the

convection loop (including the heat exchanger) the properties of the fluids

involved and the rate of heat transfer.

For the purposes of the design examples in this module, and many practical

cases, it is assumed that the maximum allowable pressure drops are given to

the designer. The designer must then predict the pressure drop for candidate

heat exchanger designs. If the pressure drop on one or both sides of the heat

exchanger is excessive, then the design is unacceptable and an alternative has

to be developed. If the pressure drops are significantly below the permissible

level then the designer may wish to attempt to reduce the size and cost of

the heat exchanger while “using” the available pressure drops.

4.2 Pressure Drop in Channels

4.2

The pressure gradient for a fluid flowing in the z direction along a channel is

given by:

dzdp

dzdp

dzdp

dzdp haf ++= (4.1)

where:

dzdp

= Pressure gradient at position z in the channel

dzdp f = frictional pressure gradient at position z in the channel

dzdpa = Pressure gradient due to the momentum change at position z

in the channel

dzdph =Hydrostatic pressure gradient at position z in the channel

and z is the coordinate in the flow direction along the channel

4.2.1 Single Phase Pressure Drop in Channels

In most single phase flows in channels (the exception being gases undergoing

significant temperature change) the pressure gradient due to momentum

change may be neglected.

With reference to Fig 4.1, the hydrostatic pressure gradient is given by:

θsinρgdzdph −= (4.2)

θ

Flow Direction

z

l

Figure 4.1 Nomenclature used in defining pressure drop

4.3

For θ = 0, i.e. a horizontal channel, then the hydrostatic pressure gradient is

zero.

For constant fluid density, or where the density change is small and a

representative mean density may be used, equation 4.1 may be integrated

over the length of the channel, and arranging the signs so that ∆p, the

pressure drop, is positive, this gives:

(4.3)

The frict

θsinglρph =∆

ional pressure gradient may be determined from:

eef

f VfVcdp ρ1ρ2 −=−= (4.4)

dddz

22

2

where c and f are the Fanning skin friction coefficient and Darcy friction

factor, as defined in equation 2.59, respectively. The hydraulic diameter, d , is

defined in equation 2.60. the negative sign in equation 4.4 indicating that the

pressure decreases in the direction of flow.

If fluid properties may be reg

4.4. may be integrated, again with the pressure drop regarded as positive:

f

e

arded as constant over a length l then equation

22 1 ll ρ2

ρ2 Vd

fVd

cpee

ff ==∆ (4.5)

Clearly, application of equation 4.4 or 4.5 requires knowledge of the

y friction factor, f, will be used

(also, to add to the potential confusion, the value

appropriate value of the factor cf or f. Since f =4cf, by definition, there is little

to choose between the two forms of equation 4.4. The student must,

however, be sure which factor is given by a particular data source.

For the remainder of this section the Darc

22o

in some texts as the Fanning friction factor and given the symbol f !)

1 Vρτ is referred to

The value of f is a function of the flow Reyno mber the r ghnes

the channel surface and the channel geometry. It will not surprise the reader

lds nu , ou s of

4.4

to learn that there are numerous correlations which may be used in the

estimation of f. As with heat transfer, the pressure drop characteristics differ

greatly depending whether the flow is laminar or turbulent, with transition

occurring at a Reynolds number of 2000-10000.

For laminar flow f is independent of surface r and inversely

oughness

proportional to the Reynolds number. Values of the constant of

proportionality for a range of channel shapes are given in Table 2.2.

For round tubes:

64Re

f = (4.6)

The simplest expression for friction factor f in turbulent flow, which is

applicable to smooth pipes, is that due to Blasius:

50.25 (3000 Re 10 )

Ref = < < (4.7a)

which may be extended to higher Reynolds numbers

0.3164

5 60.2210.0032 (10 Re 3 x 10 )f = + < < (4.7b)0.237Re

an alternative expression for commercial pipe or slightly corroded tubes:

0.42

1.0560.014Re

f = + (3380<Re) (4.7c)

44!) presented on a Moody

Diagram, as reproduced in Figure 4.2.1 Roughness values for a range of pipe

materials and conditions are given in Table 4.1.

The variation of f is traditionally (at least since 19

1 ASHRAE Handbook of Fundamentals, ASHRAE, 1997

4.5

Many heat exchanger tubes are drawn copper and therefore have a

representative roughness of some 0.0025mm; for a 19mm diameter tube this

implies a relative roughness of 0.00014, suggesting, in conjunction with figure

4.2, that equation 4.6 will give reasonable results within its range of

applicability. Steel tubes having a representative roughness of 0.025mm, for a

19mm tube this gives a relative roughness of 0.0014, rising by a further factor

of 10 when a coating of light rust forms. If dealing with initially rough tubes,

tubes which are roughened by corrosion, or high Reynolds number flow, then

the roughness must be taken into account. When carrying out hand

calculations involving rough tubes or pipes then the quickest method of

estimating f is to use a Moody Diagram.

4.6

Table 4.1 Roughness value ε

or calculations using a computer it is necessary to put this data in numerical

rm. Moody produced a correlation (equation 4.8) which matched his

iagram to within 5% for Reynolds numbers between 4000 and 107 and for

alue

F

fo

d

s of ε/d of less than 0.01. v

0.336100.005496 1 20000Re

fdε⎡ ⎤⎛ ⎞

⎢ ⎥= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Alternative correlations are available, for example ASHRAE recommend that

gure 4.2, the friction

factor becomes independent of Reynolds number, the

(4.8)

2

for complete turbulence, where, as can be seen from fi

following is used:

101 1.14 2log df ε

⎛ ⎞= + ⎜ ⎟⎝ ⎠

(4.9)

In general,

2 ASHRAE Handbook of Fundamentals, ASHRAE, 1997

4.7

( )10 101 9.31.14 2log 2log 1

Red

f d fε ε

⎛ ⎞⎛ ⎞= + − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ (4.10)

The difficulty with this equation being t

nal pressure gradient or drop in long, non-circular channels

(including annuli) may be estimated by using equation 4.4 or 4.5 and

evaluating f for laminar flow using table 2.2. or for turbulent flow using one

of equations 4.5-4.10.

Entry effects are significant when the channels are short, for example in many

h are appropriate to plate-heat exchangers, however

the general procedure for calculating the pressure drop remains the same

hat it must be solved iteratively.

The frictio

compact heat exchangers or in heat sinks for cooling electronic devices, then

this must be taken into account. Values of f or cf are available for many

plate-fin surfaces. Examples are given in figure 2.16. Correlations are available

for friction factors whic

and is based upon equation 4.4.

4.8

4.2.2 Single phase pressure drop across tube bundles

The pressure drop across a tube bundle depends on the geometry of the

bundle, fluid properties and flow rate. For plain tube bundles, assuming

constant fluid properties, the pressure drop (excluding the gravitational

component) is given by

2

maxEu2 r

Vp Nρ∆ = (4.11)

where Eu is the Euler number and Nr is the number of tube rows. Eu is

analogous to the friction factor in internal flows (and is referred to as a

friction factor in some texts). Vmax is the maximum velocity between the

tubes, as determined from figure .3.12 and equation 3.84 or 3.85.

(Re , , ,dEu f a b N= )r (4.12)

where (using the nomenclature of fig 3.12), , TS Sa bd d

= = L

. For many tube

rows, Eu is independent of Nr.

Curves showing Eu/k1 for inline and staggered tube banks having a large

number of rows with (k1 being unity for a=b) together with values of k1 for

are reproduced here as figures 4.3(a) and (b)a b≠ 3. Equations have been

fitted to these curves, However because of the complexity of the

relationships these are not reproduced here. They are available in 4, where 2

and 3 Re ranges, for each value of a, with appropriate equations to correlate

Eu and k1 for in-line and staggered tube banks respectively.

The pressure drop in the first few (3 or 4) rows differs from that predicted

from fig 4.3. It may be higher or lower than the average value, depending

upon geometry and Re. Correction factors may be defined

3 Handbook of Heat Exchanger Design, Ed Hewitt, G.F., Begell House, New York, 1992 4 Handbook of Heat Exchanger Design, Ed Hewitt, G.F., Begell House, New York, 1992

4.9

* / and /z z z zc Eu Eu C Eu Eu= = where Eu is the Euler number from figure

4.3, *zEu is the Euler number determined from the pressure drop over tube

row z and Euz is the Euler number relating to the pressure drop for a bank

having z rows. Table 4.2 gives values of cz and Cz for bundles having less than

10 rows.

Figure 4.3a Pressure drop of in-line banks as referred to the relative longitudinal pitch b

Figure 4.3b Pressure drop of staggered banks as referred to the relative transverse pitch a

4.10

Table 4.2 Correction factors for row-to-row variations

4.2.3 Property variations

In general, fluid properties should be evaluated at the mean bulk temperature

When the fluid properties vary due to heat transfer (or in the case of a gas,

due to pressure drop) this may require the introduction of correction

factors, or in the case of properties varying in the flow direction, division of

the heat exchanger into several sections over which the fluid properties may

be regarded as constant.

If the temperature difference between the wall and the bulk of the fluid is

large, the viscosity may vary significantly between the bulk of the fluid and the

fluid close to the wall. Typically, a correction factor of the form n

w

bulk

µη

⎛ ⎞⎜ ⎟⎝ ⎠

is

used.

4.2.4 Pressure drop in nozzles and headers.

The pressure drop in headers and nozzles (and in pipe fittings in general) is

usually expressed in terms of velocity heads. The appropriate velocities and

typical values of K, the number of velocity heads lost, are given below.

4.11

Channel (i.e. tube-side) inlet and outlet nozzles:

2 2

2 2n

nt nt nt

G Vp K K

ρρ

⎛ ⎞ ⎛∆ = =⎜ ⎟ ⎜⎜ ⎟ ⎜

⎝ ⎠ ⎝

n⎞⎟⎟⎠ (4.13)

based on mass flux or velocity in nozzles.

1.1 (inlet nozzle)0.7 (outlet nozzle)

ntK ==

Headers:

2 2

2 2t t

h h p nt

G Vpp K N K ρ

ρ

⎛ ⎞ ⎛ ⎞∆ = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

N (4.14)

based on mass flux or velocity in tubes, Np= number of tube side passes.

0.9 (one tube-side pass)1.6 (two or more tube side passes)

kK ==

Shell- side inlet and outlet nozzles:

2 2

2 2n

ns ns ns

G Vp K K

ρρ

⎛ ⎞ ⎛∆ = =⎜ ⎟ ⎜⎜ ⎟ ⎜

⎝ ⎠ ⎝

n⎞⎟⎟⎠

(4.15)

based on mass flux or velocity in nozzles.

With impingement plate 1.0 nnsi

e

AKA

⎛ ⎞= + ⎜ ⎟

⎝ ⎠

Without impingement plate :( ) ( ){ }2

11.00.6

nsi

e n o

KA A S d S

⎛ ⎞⎜ ⎟= +⎜ ⎟+ −⎝ ⎠

Where

An= flow area of nozzle

Ae= escape are at nozzle = (perimeter of nozzle x distance from

nozzle to impingement plate or closest tubes)

do= tube outside diameter

S=tube pitch

4.12

Nozzle and header losses must be added to the pressure drops calculated for

the core of the heat exchanger.

4.3 Two-phase pressure drop (vapour + liquid)

The evaluation of the pressure drop during two-phase flow in a heat

exchanger is generally complex. Each component of the pressure gradient is a

function of, amongst other parameters, the quality, x, defined as the ratio of

vapour mass flow to total mass flow. Each component of pressure gradient

listed in equation 4.1 must be evaluated independently and applied to a short

flow length, using a mean value of x for that length. The pressure drop for

that length is then determined. The mean quality for the next increment of

flow length is then calculated and the process repeated. The pressure drop is

then the sum of the pressure gradient x section length for each of the

sections. A comprehensive guide to the correlations which may be used is

given in5

5 Handbook of Heat Exchanger Design, Chapter 2.3, Ed Hewitt, G.F., Begell House, New York, 1992

4.13

Summary Points

• The acceptable pressure drop in a heat exchanger is usually specified.

• The designer should ensure the allowable pressure drop is not exceeded, but in

optimising a design should attempt to 'use' all of the available pressure drop.

• Correlations are available which allow the pressure drop in the heat exchanger

core and attachments to be calculated.

• Pressure drops in single-phase, incompressible flows are relatively easy to

calculate. For two-phase flows numerical integration is required, a computer

package or programme is required for all but the simplest of cases.

4.14