Heat Exchangers

10/22/2015 18.5 Heat Exchangers

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Thermodynamics and Propulsion

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Subsections

18.5.1 Simplified Counterflow Heat Exchanger (With Uniform Wall Temperature)18.5.2 General Counterflow Heat Exchanger18.5.3 Efficiency of a Counterflow Heat Exchanger

18.5 Heat ExchangersThe general function of a heat exchanger is to transfer heat from one fluid to another. The basiccomponent of a heat exchanger can be viewed as a tube with one fluid running through it and anotherfluid flowing by on the outside. There are thus three heat transfer operations that need to bedescribed:

1. Convective heat transfer from fluid to the inner wall of the tube,2. Conductive heat transfer through the tube wall, and3. Convective heat transfer from the outer tube wall to the outside fluid.

Heat exchangers are typically classified according to flow arrangement and type of construction. Thesimplest heat exchanger is one for which the hot and cold fluids move in the same or oppositedirections in a concentric tube (or doublepipe) construction. In the parallelflow arrangement ofFigure 18.8(a), the hot and cold fluids enter at the same end, flow in the same direction, and leave atthe same end. In the counterflow arrangement of Figure 18.8(b), the fluids enter at opposite ends, flowin opposite directions, and leave at opposite ends.

Figure 18.8: Concentric tubes heat exchangers[Parallel flow] [Counterflow]

[Finned with both fluids unmixed.] [Unfinned with one fluid mixed

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Figure 18.9: Crossflow heat exchangers.and the other unmixed]

Alternatively, the fluids may be in cross flow (perpendicular to each other), as shown by the finned andunfinned tubular heat exchangers of Figure 18.9. The two configurations differ according to whetherthe fluid moving over the tubes is unmixed or mixed. In Figure 18.9(a), the fluid is said to be unmixedbecause the fins prevent motion in a direction ( ) that is transverse to the main flow direction ( ). In

this case the fluid temperature varies with and . In contrast, for the unfinned tube bundle of

Figure 18.9(b), fluid motion, hence mixing, in the transverse direction is possible, and temperaturevariations are primarily in the main flow direction. Since the tube flow is unmixed, both fluids areunmixed in the finned exchanger, while one fluid is mixed and the other unmixed in the unfinnedexchanger.

To develop the methodology for heat exchanger analysis and design, we look at the problem of heattransfer from a fluid inside a tube to another fluid outside.

Figure 18.10: Geometry for heattransfer between two fluids

We examined this problem before in Section 17.2 and found that the heat transfer rate per unit lengthis given by

(18..21)

Here we have taken into account one additional thermal resistance than in Section 17.2, theresistance due to convection on the interior, and include in our expression for heat transfer the bulktemperature of the fluid, , rather than the interior wall temperature, .

It is useful to define an overall heat transfer coefficient per unit length as

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(18..22)

From (18.21) and (18.22) the overall heat transfer coefficient, , is

(18..23)

We will make use of this in what follows.

Figure 18.11: Counterflow heat exchanger

A schematic of a counterflow heat exchanger is shown in Figure 18.11. We wish to know thetemperature distribution along the tube and the amount of heat transferred.

18.5.1 Simplified Counterflow Heat Exchanger (With UniformWall Temperature)To address this we start by considering the general case of axial variation of temperature in a tubewith wall at uniform temperature and a fluid flowing inside the tube (Figure 18.12).

Figure 18.12: Fluid temperature distribution along thetube with uniform wall temperature

The objective is to find the mean temperature of the fluid at , , in the case where fluid comes

in at with temperature and leaves at with temperature . The expected



distribution for heating and cooling are sketched in Figure 18.12.

For heating ( ), the heat flow from the pipe wall in a length is

where is the pipe diameter. The heat given to the fluid (the change in enthalpy) is given by

where is the density of the fluid, is the mean velocity of the fluid, is the specific heat of the

fluid and is the mass flow rate of the fluid. Setting the last two expressions equal and integratingfrom the start of the pipe, we find

Carrying out the integration,

i.e.,

(18..24)

Equation (18.24) can be written as

where

This is the temperature distribution along the pipe. The exit temperature at is

(18..25)

The total heat transfer to the wall all along the pipe is



(18..26)

From Equation (18.25),

The total rate of heat transfer is therefore

or

(18..27)

where is the logarithmic mean temperature difference, defined as

(18..28)

The concept of a logarithmic mean temperature difference is useful in the analysis of heat exchangers.We will define a logarithmic mean temperature difference for the general counterflow heat exchangerbelow.

18.5.2 General Counterflow Heat ExchangerWe return to our original problem, to Figure 18.11, and write an overall heat balance between the twocounterflowing streams as

From a local heat balance, the heat given up by stream in length x is . (There is a

negative sign since decreases). The heat taken up by stream is . (There is a

negative sign because decreases as increases). The local heat balance is



(18..29)

Solving (18.29) for and , we find

where . Also, where is the overall heat transfer coefficient. We can then

say

Integrating from to gives

(18..30)

Equation (18.30) can also be written as

(18..31)

where

We know that

(18..32)

Thus

Solving for the total heat transfer:



(18..33)

Rearranging (18.30) allows us to express in terms of other parameters as

(18..34)

Substituting (18.34) into (18.33) we obtain a final expression for the total heat transfer for acounterflow heat exchanger:

(18..35)

or

(18..36)

This is the generalization (for nonuniform wall temperature) of our result from Section 18.5.1.

18.5.3 Efficiency of a Counterflow Heat Exchanger

Suppose we know only the two inlet temperatures , , and we need to find the outlet

temperatures. From (18.31),

or, rearranging,

(18..37)



Eliminating from (18.32),

(18..38)

We now have two equations, (18.37) and (18.38), and two unknowns, and . Solving first for

,

or

(18..39)

where is the efficiency of a counterflow heat exchanger:

(18..40)

Equation 18.39 gives in terms of known quantities. We can use this result in (18.38) to find :

We examine three examples.

1.

can approach zero at cold end.

as , surface area, .

Maximum value of ratio



Maximum value of ratio .

2.

is negative, as

Maximum value of ratio

Maximum value of ratio .

3.

temperature difference remains uniform, .

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Heat Exchangers

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