A Model of A Simple Heat Exchanger
A simple heat exchanger might be thought of as two straight pipes with fluid flow, which are thermally connected. Let the pipes be of equal length L, carrying fluids with heat capacity (energy per unit mass per unit change in temperature) and let the mass flow rate of the fluids through the pipes be (mass per unit time), where the subscript i applies to pipe 1 or pipe 2.
Temperature profiles for the pipes are and where x is the distance along the pipe. Assume a steady state, so that the temperature profiles are not functions of time. Assume also that the only transfer of heat from a small volume of fluid in one pipe is to the fluid element in the other pipe at the same position. There is no transfer of heat along a pipe due to temperature differences in that pipe. By Newton's law of cooling the rate of change in energy of a small volume of fluid is proportional to the difference in temperatures between it and the corresponding element in the other pipe:
where is the thermal energy per unit length and γ is the thermal connection constant per unit length between the two pipes. This change in internal energy results in a change in the temperature of the fluid element. The time rate of change for the fluid element being carried along by the flow is:
where is the "thermal mass flow rate". The differential equations governing the heat exchanger may now be written as:
Note that, since the system is in a steady state, there are no partial derivatives of temperature with respect to time, and since there is no heat transfer along the pipe, there are no second derivatives in x as is found in the heat equation. These two coupled first-order differential equations may be solved to yield:
where, and A and B are two as yet undetermined constants of integration. Let and be the temperatures at x=0 and let and be the temperatures at the end of the pipe at x=L. Define the average temperatures in each pipe as:
Using the solutions above, these temperatures are:
Choosing any two of the temperatures above eliminates the constants of integration, letting us find the other four temperatures. We find the total energy transferred by integrating the expressions for the time rate of change of internal energy per unit length:
By the conservation of energy, the sum of the two energies is zero. The quantity is known as the Log mean temperature difference, and is a measure of the effectiveness of the heat exchanger in transferring heat energy.
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