The Overall Heat Transfer...

The Overall Heat Transfer Coefficient

• A heat exchanger typically involves two flowing fluids

separated by a solid wall. Heat is first transferred from

the hot fluid to the wall by convection, through the wall

by conduction, and from the wall to the cold fluid by

convection. Any radiation effects are usually included in

the convection heat transfer coefficients.

• The thermal resistance network associated with this heat

transfer process involves two convection resistances and

one conduction resistance (Fig. 8). The subscripts i and o

denote the inner and outer surfaces of the inner tube.

1

Overall Heat Transfer Coefficient continued

• For a double-pipe heat exchanger, the area of the inner

surface of the wall that separates the two fluids (i.e. the

surface area of the separating wall wetted by the inner

fluid) is Ai = πDiL, the area of the outer surface of the

wall (i.e. the surface area of the separating wall wetted

by the outer fluid) is Ao = πDoL, and the thermal

resistance of the tube wall in this case is: -

• Then the total thermal resistance becomes: -

2

3

Fig. 8: Thermal resistance network associated with

heat transfer in a double-pipe heat exchanger


• Combine all the thermal resistances in the path of heat

flow from the hot fluid to the cold into a single

resistance R, the rate of heat transfer between the two

fluids can be expressed as: -

• Where; U is the overall heat transfer coefficient

(W/m2·°C) and As is the heat transfer surface area (m2).

• Cancelling ΔT, equation 3 reduces to: -

4


• There are two overall heat transfer coefficients Ui and Uo

for a heat exchanger because every heat exchanger has

two heat transfer surface areas Ai and Ao, which in

general are not equal to each other. That is; UiAi = UoAo,

and Ui ≠ Uo unless Ai = Ao. Therefore, the overall heat

transfer coefficient U of a heat exchanger is meaningless

unless the area on which it is based is specified.

• When the wall thickness of the tube is small or the

thermal conductivity of the tube material is high, the

thermal resistance of the tube is negligible (Rwall = 0),

and the inner and outer surfaces of the tube are almost

identical (i.e. Ai = Ao = As).

5


• Thus equation 4 for the overall heat transfer coefficient simplifies to: -

• Where U = Ui = Uo, and hi and ho, are the individual convection heat transfer coefficients inside and outside the tube, which are determined using convection relations.

• Since the inverse of a large number is small, the overall heat transfer coefficient U in equation 5 is dominated by the smaller convection coefficient. When one of the convection coefficients is much smaller than the other (say, hi ˂˂ ho), 1/hi ˃˃ 1/ho, and thus U ≈ hi).

6


• Therefore, the smaller heat transfer coefficient creates a

bottleneck on the path of heat flow and seriously

impedes heat transfer. This situation arises frequently

when one of the fluids is a gas and the other is a liquid.

In such cases, fins are commonly used on the gas side to

enhance the product UAs and thus the heat transfer on

that side.

• Representative values of the overall heat transfer

coefficient U are given in Table 1. The overall heat

transfer coefficient ranges from about 10 W/m2·°C for

gas-to-gas heat exchangers to about 10,000 W/m2·°C for

heat exchangers that involve phase changes.

7

8

Type of heat exchanger U (W/m2·°C)

Water-to-water 850–1700

Water-to-oil 100–350

Water-to-gasoline or kerosene 300–1000

Feed water heaters 1000–8500

Steam-to-light fuel oil 200–400

Steam-to-heavy fuel oil 50–200

Steam condenser 1000–6000

Freon condenser (water cooled) 300–1000

Ammonia condenser (water cooled) 800–1400

Alcohol condensers (water cooled) 250–700

Gas-to-gas 10–40

Water-to-air in finned tubes (water in tubes) 30–60 Based on air-side surface area

400–850 Based on water-side surface area

Steam-to-air in finned tubes (steam in tubes) 30–300 Based on air-side surface area

400–4000 Based on steam-side surface area

Table 1: Representative values of the overall heat

transfer coefficients in heat exchangers


• When the tube is finned on one side to enhance heat

transfer, the total heat transfer surface area on the finned

side becomes: -

• Where; Afin is the surface area of the fins and Aunfinned is

the area of the unfinned portion of the tube surface.

• For short fins of high thermal conductivity, this total area

can be used in the convection resistance relation Rconv =

1/hAs since the fins in this case will be very nearly

isothermal.

9


• Otherwise, the effective surface area A should be

determined from: -

• Where; ηfin is the fin efficiency.

• For isothermal fins ηfin = 1, and thus equation 7 reduces

to equation 6.

• Experience shows that convection heat transfer strongly

depends on the fluid properties (i.e. dynamic viscosity μ,

thermal conductivity K, density ρ, and specific heat Cp,

as well as the fluid velocity V).

10


• It also depends on the geometry and the roughness of the

solid surface, in addition to the type of fluid flow (such

as being streamlined (laminar) or turbulent). Thus,

convection heat transfer relations will be rather complex

because of the dependence of convection on so many

variables.

• Despite the complexity of convection, the rate of

convection heat transfer is observed to be proportional to

the temperature difference and is conveniently expressed

by Newton’s law of cooling as: -

11


• Where; h is the convection heat transfer coefficient

defined as the rate of heat transfer between a solid

surface and a fluid per unit surface area per unit

temperature difference (W/m2.°C), As the heat transfer

surface area (m2), Ts the temperature of the surface (°C),

and T∞ the temperature of the fluid sufficiently far from

the surface (°C).

• Regardless of the simple appearance of equation 8, the

convection heat transfer coefficient h is difficult to

determine as convection heat transfer relations are rather

complex because of its dependence on the several

mentioned variables.

12


• Flow in a tube can be laminar or turbulent, depending on

the flow conditions. Fluid flow is streamlined (i.e.

Laminar) at low velocities, but turns turbulent as the

velocity is increased beyond a critical value. Laminar

flow is encountered when highly viscous fluids such as

oils flow in small diameter tubes or narrow passages. For

flow in a circular tube, the Reynolds number is defined

as: -

• Where; Vm is the mean fluid velocity, D the diameter of

the tube, and ν = μ/ρ the kinematic viscosity of the fluid.

13


• Under most practical conditions, the flow in a tube is laminar for Re ˂ 2300, turbulent for Re ˃ 10000 and transitional in between.

• For flow through noncircular tubes, the Reynolds number as well as the Nusselt number and the friction factor are based on the hydraulic diameter Dh defined as: -

• Where; Ac is the cross sectional area of the tube and p is its perimeter. For an annulus the hydraulic diameter of annulus given by: -

14


• Where; Di and Do are the inner and outer diameters of the concentric annulus.

• The hydraulic diameter is defined such that it reduces to ordinary diameter D for circular tubes since: -

• Nusselt number is another very important group used with convection. It is a non-dimensional variable defined as: -

• Where; Kfluid is the thermal conductivity of the fluid and Lc is the characteristic length.

15


• The Nusselt number represents the enhancement of heat

transfer through a fluid layer as a result of convection

relative to conduction across the same fluid layer. A

Nusselt number of Nu = 1 for a fluid layer represents

heat transfer across the layer by pure conduction.

• Annular flow may involve heat transfer on both inner

and outer surfaces and therefore is associated with two

Nusselt numbers: Nui on the inner tube surface and Nuo

on the outer tube surface. The Nusselt numbers for fully

developed laminar flow with one surface isothermal and

the other adiabatic are given in Table 2.

16

17

Di/Do Nui Nuo

0 - 3.66

0.05 17.46 4.06

0.10 11.56 4.11

0.25 7.37 4.23

0.50 5.74 4.43

1.00 4.86 4.86

Table 2: Nusselt number for fully developed laminar flow

in an annulus with one surface isothermal and the other adiabatic


• When Nusselt numbers are known, the convection coefficients for the inner and the outer surfaces are determined from: -

• Where; Di, Do, and Dh are the inner, outer and hydraulic diameters of the concentric annulus.

• For fully developed turbulent flow, the inner and outer convection coefficients are approximately equal to each other, and the tube annulus can be treated as a noncircular duct with a hydraulic diameter Dh = Do - Di. The Nusselt number in this case can be determined from a suitable turbulent flow relation.

18


• Prandtl Number

• The thickness of the thermal boundary layer increases in the flow direction, since the effects of heat transfer are felt at greater distances from the surface further downstream.

• Because the convection heat transfer rate anywhere along the surface is directly related to the temperature gradient at that location, the shape of the temperature profile in the thermal boundary layer dictates the convection heat transfer between a solid surface and the fluid flowing over it. In flow over a heated or cooled surface, both velocity and thermal boundary layers will develop simultaneously.

19


• As the fluid velocity will have a strong influence on the

temperature profile, the development of the velocity

boundary layer relative to the thermal boundary layer

will have a strong effect on the convection heat transfer.

• The relative thickness of the velocity and the thermal

boundary layers is best described by the dimensionless

parameter Prandtl number defined as: -

• The Prandtl numbers of fluids range from less than 0.01

for liquid metals to more than 100,000 for heavy oils.

20


• The Prandtl number of water is in the order of 10. The

Prandtl numbers of gases are about 1 (i.e. both

momentum and heat dissipate through the fluid at about

the same rate).

• Heat diffuses very quickly in liquid metals (Pr ˂˂ 1) and

very slowly in oils (Pr ˃˃ 1) relative to momentum.

Consequently the thermal boundary layer is much

thicker for liquid metals and much thinner for oils

relative to the velocity boundary layer.

21

22

Fig. 9: The development of the velocity and thermal

boundary layers for flow over a surface

V

0.99V

Zero slip velocity

y 0

V

y

y

s0.99 T T

t y 0

T

y

y


• When a fluid is forced to flow over a solid nonporous

surface (i.e. impermeable to the fluid), the fluid in

motion comes to a complete stop at the surface and

assumes a zero velocity relative to the surface (i.e. the

fluid layer in direct contact with the solid surface sticks

to the surface and there is no slip). This phenomenon,

which is due to the viscosity of the fluid, is known as the

no-slip condition. The no-slip condition is responsible

for the development of the velocity profile for flow (i.e.

because of friction between the fluid layers, the layer

that sticks to the wall slows the adjacent fluid layer,

which slows the next layer, and so on).

23


• A consequence of the no-slip condition is that all velocity profiles must have zero values at the points of contact between a fluid and a solid. From heat transfer point of view, the indication of no-slip velocity is that the mechanism of heat transfer from the solid surface to the fluid layer adjacent to the surface is by pure conduction since the fluid layer is motionless. The conducted heat is then convected away from the surface as a result of fluid motion. This can be expressed as: -

• Where; T represents the temperature distribution in the fluid and (∂T/∂y)y=0 is the temperature gradient at the surface.

24


• Equate equations 8 and 16 for the heat flux (i.e. the rate

of heat transfer per unit time per unit surface area) yields

the convection heat transfer coefficient: -

• Equation 17 indicates that the heat transfer coefficient h

can be determined if we have the temperature gradient

very close to the wall.

• Although experiments can be done to measure the

temperature gradient, yet theses experiments have to be

controlled very carefully.

25


• In general, it must be realised that the heat transfer

coefficient is not constant; it is actually a function of

position (i.e. varies along the flow direction due to the

change in the temperature gradient with position).

• In general, except when the flow is fully developed, the

heat transfer coefficients are not the same at the different

positions. Thus in many times, an average value will be

used. The average or mean convection heat transfer

coefficient for a surface is determined by properly

averaging the local convection heat transfer coefficients

over the entire surface.

26


• Thus in many times, an average value will be used. The

average or mean convection heat transfer coefficient for

a surface is determined by properly averaging the local

convection heat transfer coefficients over the entire

surface.

27

Fouling Factor

• The performance of heat exchangers usually deteriorates

with time as a result of accumulation of deposits on heat

transfer surfaces. The layer of deposits represents

additional resistance to heat transfer and causes the rate

of heat transfer in a heat exchanger to decrease. The net

effect of these accumulations on heat transfer is

represented by a fouling factor Rf, which is a measure of

the thermal resistance introduced by fouling.

• The most common type of fouling is the precipitation of

solid deposits in a fluid on the heat transfer surfaces.

28

Fouling Factor

• This type of fouling can be observed by noticing a layer of calcium-based deposits on the inner surfaces a teapot (the surfaces at which boiling occurs) after prolonged use especially in areas where the water is hard. Mineral deposits forming on the inner surfaces of fine tubes in a heat exchanger (Fig. 10) is another example. The scales of such deposits come off by scratching, and the surfaces can be cleaned of such deposits by chemical treatment.

• To avoid the detrimental effect these deposits may have on the flow passage area and the heat transfer, water in power and process plants is extensively treated and its solid contents are removed before it is allowed to circulate through the system.

29

30

Fig. 10: Precipitation fouling of ash particles on super heater tubes

Fouling Factor continued

• The solid ash particles in the flue gases accumulating on

the surfaces of air pre-heaters create similar problems.

• Another form of fouling, which is common in chemical

process industry, is corrosion and other chemical fouling

in which the surfaces are fouled by the accumulation of

the products of chemical reactions on the surfaces.

Corrosion fouling can be avoided by coating metal pipes

with glass or using plastic pipes instead of metal ones.

• Heat exchangers may also be fouled by the growth of

algae (biological fouling) in warm fluids and can be

prevented by chemical treatment.

31


• Fouling should be considered in the design and selection of heat exchangers in applications where it is likely to occur (i.e. select a larger and thus more expensive heat exchanger to ensure that it meets the design heat transfer requirements even after fouling occurs).

• The periodic cleaning of heat exchangers and the resulting down time are additional penalties associated with fouling.

• The fouling factor is obviously zero for a new heat exchanger and increases with time as the solid deposits build up on the heat exchanger surface. The fouling factor depends on the operating temperature and the velocity of the fluids, as well as the length of service.

32


• Fouling increases with increasing temperature and

decreasing velocity. The overall heat transfer coefficient

relation given equation 4 is valid for clean surfaces and

must be modified to account for the effects of fouling on

both the inner and the outer surfaces of the tube. For an

unfinned shell-and-tube heat exchanger, it can be

expressed as: -

• Where; Ai = πDiL and Ao = πDoL are the areas of inner

and outer surfaces, and Rf, i and Rf, o are the fouling

factors at those surfaces.

33

34

Fluid Rf (m2·°C/W)

Distilled water, sea water, river water, boiler feed water 0.0001 Below 50°C

0.0002 Above 50°C

Fuel oil 0.0009

Steam (oil-free) 0.0001

Liquid Refrigerants 0.0002

Vapour Refrigerants 0.0004

Alcohol vapours 0.0001

Air 0.0004

Table 3: Representative fouling factors

(thermal resistance due to fouling for a unit surface area)


• Representative values of fouling factors are given in

Table 3. More comprehensive tables of fouling factors

are available in handbooks. Considerable uncertainty

exists in representative values of fouling factors available

in handbooks, and they should be used as a guide in

selection and evaluation of heat exchangers to account

for the effects of anticipated fouling on heat transfer.

• Most fouling factors are of the order of 10-4 m2·°C/W,

which is equivalent to the thermal resistance of 0.2 mm

thick limestone layer (K = 2.9 W/m·°C) per unit surface

area.

35


• Therefore, in the absence of specific data, the surfaces

can be assumed to be coated with 0.2 mm of limestone

as a starting point to account for the effects of fouling.

36

Example

• Hot oil is to be cooled in a double-tube counter-flow heat exchanger. The copper inner tubes have a diameter of 2 cm and negligible thickness. The inner diameter of the outer tube (the shell) is 3 cm. Water flows through the tube at a rate of 0.5 kg/s, and the oil through the shell at a rate of 0.8 kg/s. Taking the average temperatures of the water and the oil to be 45°C and 80°C, respectively, determine the overall heat transfer coefficient of this heat exchanger assuming that the thermal resistance of the inner tube is negligible since the tube material is highly conductive and its thickness is negligible, both the oil and water flow are fully developed, and that properties of the oil and water are constant.

37

Example Solution

• The properties of water at 45°C are ρ = 990 kg/m3, Pr =

3.91, K = 0.637 W/m·°C, and ν = μ/ρ = 0.602x10-6 m2/s.

• The properties of oil at 80°C are ρ = 852 kg/m3, Pr =

490, K = 0.138 W/m·°C, and ν = μ/ρ = 37.5x10-6 m2/s.

• The schematic of the heat exchanger is given in the

Figure above. The overall heat transfer coefficient U can

be determined from equation 5 as: -

• Where; hi and ho are the convection heat transfer

coefficients inside and outside the tube respectively,

which are to be determined using convection relations.

39

Example Solution continued

• The hydraulic diameter for a circular tube is the diameter

of the tube itself, Dh = D = 0.02 m. The mean velocity of

water in the tube and the Reynolds number are: -

• Therefore, the flow of water is turbulent. Assuming the

flow to be fully developed, the Nusselt number can be

determined from Dittus-Boelter equation: -

40


• Alternatively, the Nusselt number can be determined

from Gnielinski correlation: -

• Where f is the Darcy friction factor that can be obtained

for smooth tubes from Petukhov correlation by: -

41


• Repeat the analysis above for oil, the hydraulic diameter

for the annular space is Dh = Do - Di = 0.03 - 0.02 = 0.01

m. The mean velocity and the Reynolds number in this

case are: -

• Assuming fully developed flow, the Nusselt number on

the tube side of the annular space Nui corresponding to

Di/Do = 0.02/0.03 = 0.667 can be determined from Table

2 by interpolation as Nui = 5.45.

42


• Thus: -

• Then the overall heat transfer coefficient for this heat

exchanger becomes: -

43

The Overall Heat Transfer...

Documents

Transcript of The Overall Heat Transfer...