|Thermodynamics and Propulsion|
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 Figure 18.8(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 Figure 18.8(b), the fluids enter at opposite ends, flow in opposite directions, and leave at opposite ends.
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
Here we have taken into account one additional thermal resistance than in Section 17.2, 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
From (18.21) and (18.22) the overall heat transfer coefficient, , is
We will make use of this in what follows.
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.
and a fluid flowing inside the tube (Figure 18.12).
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,
Equation (18.24) can be written as
This is the temperature distribution along the pipe. The exit temperature at is
The total heat transfer to the wall all along the pipe is
From Equation (18.25),
The total rate of heat transfer is therefore
where is the logarithmic mean temperature difference, defined as
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.
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
Solving (18.29) for and , we find
where . Also, where is the overall heat transfer coefficient. We can then say
Integrating from to gives
Equation (18.30) can also be written as
We know that
Solving for the total heat transfer:
Rearranging (18.30) allows us to express in terms of other parameters as
Substituting (18.34) into (18.33) we obtain a final expression for the total heat transfer for a counterflow heat exchanger:
This is the generalization (for non-uniform wall temperature) of our result from Section 18.5.1.
Suppose we know only the two inlet temperatures , , and we need to find the outlet temperatures. From (18.31),
Eliminating from (18.32),
We now have two equations, (18.37) and (18.38), and two unknowns, and . Solving first for ,
where is the efficiency of a counterflow heat exchanger:
Equation 18.39 gives in terms of known quantities. We can use this result in (18.38) to find :
We examine three examples.