Thermodynamics and Propulsion

Next: 18.6 Muddiest Points on Up: 18. Generalized Conduction and Previous: 18.4 Modeling Complex Physical Contents Index Subsections 18.5.1 Simplified Counterflow Heat Exchanger (With Uniform Wall Temperature) 18.5.2 General Counterflow Heat Exchanger 18.5.3 Efficiency of a Counterflow Heat Exchanger

18.5 Heat Exchangers The general function of a heat exchanger is to transfer heat from one fluid to another. The basic component of a heat exchanger can be viewed as a tube with one fluid running through it and another fluid flowing by on the outside. There are thus three heat transfer operations that need to be described: 1. Convective heat transfer from fluid to the inner wall of the tube, 2. Conductive heat transfer through the tube wall, and 3. 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. The simplest heat exchanger is one for which the hot and cold fluids move in the same or opposite directions in a concentric tube (or double-pipe) construction. In the parallel-flow arrangement of Figure18.8(a), the hot and cold fluids enter at the same end, flow in the same direction, and leave at the same end. In the counterflow arrangement of Figure18.8(b), the fluids enter at opposite ends, flow in opposite directions, and leave at opposite ends. [Parallel flow] [Counterflow]

Figure 18.8: Concentric tubes heat exchangers

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

Figure 18.9: Cross-flow heat exchangers.

Alternatively, the fluids may be in cross flow (perpendicular to each other), as shown by the finned and unfinned tubular heat exchangers of Figure18.9. The two configurations differ according to whether the fluid moving over the tubes is unmixed or mixed. In Figure18.9(a), the fluid is said to be unmixed because 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 Figure18.9(b), fluid motion, hence mixing, in the transverse direction is possible, and temperature variations are primarily in the main flow direction. Since the tube flow is unmixed, both fluids are unmixed in the finned exchanger, while one fluid is mixed and the other unmixed in the unfinned exchanger. To develop the methodology for heat exchanger analysis and design, we look at the problem of heat transfer from a fluid inside a tube to another fluid outside.

Figure 18.10: Geometry for heat transfer between two fluids

We examined this problem before in Section17.2 and found that the heat transfer rate per unit length is given by (18..21)

Here we have taken into account one additional thermal resistance than in Section17.2, the resistance due to convection on the interior, and include in our expression for heat transfer the bulk temperature of the fluid, , rather than the interior wall temperature, . It is useful to define an overall heat transfer coefficient per unit length as (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 Figure18.11. We wish to know the temperature distribution along the tube and the amount of heat transferred.

18.5.1 Simplified Counterflow Heat Exchanger (With Uniform Wall Temperature) To address this we start by considering the general case of axial variation of temperature in a tube with wall at uniform temperature and a fluid flowing inside the tube (Figure18.12).

Figure 18.12: Fluid temperature distribution along the tube 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 Figure18.12. 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 integrating from 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 exchanger below. 18.5.2 General Counterflow Heat Exchanger We return to our original problem, to Figure18.11, and write an overall heat balance between the two counterflowing 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 a counterflow heat exchanger: (18..35)

or

(18..36)

This is the generalization (for non-uniform wall temperature) of our result from Section18.5.1.

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)

Equation18.39 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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