Thermodynamics and Propulsion  
Subsections
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:

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

Alternatively, the fluids may be in cross flow (perpendicular to each other), as shown by the finned and unfinned tubular heat exchangers of Figure 18.9. The two configurations differ according to whether the fluid moving over the tubes is unmixed or mixed. In Figure 18.9(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 Figure 18.9(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.
We examined this problem before in Section 17.2 and found that the heat transfer rate per unit length is given by
It is useful to define an overall heat transfer coefficient per unit length as
A schematic of a counterflow heat exchanger is shown in Figure 18.11. We wish to know the temperature distribution along the tube and the amount of heat transferred.
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 integrating from the start of the pipe, we find
Carrying out the integration,
i.e.,
where
This is the temperature distribution along the pipe. The exit temperature at is
The total rate of heat transfer is therefore
or
 
(18..27) 
We return to our original problem, to Figure 18.11, 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
where . Also, where is the overall heat transfer coefficient. We can then say
Integrating from to gives
We know that
Solving for the total heat transfer:
(18..35)  
or
 
(18..36) 
Suppose we know only the two inlet temperatures , , and we need to find the outlet temperatures. From (18.31),
or, rearranging,
 
(18..37) 
or
We examine three examples.
can approach zero at cold end.
as , surface area, .
Maximum value of ratio
Maximum value of ratio .
is negative, as
Maximum value of ratio
Maximum value of ratio .
temperature difference remains uniform, .
UnifiedTP