| Summary: | The transfer functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>G</mi><mo stretchy="false">^</mo></mover><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing <i>N</i> spatial sections of the exchanger, where each <i>n</i>th section can be described by a simple rational transfer function matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>G</mi><mo stretchy="false">^</mo></mover><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>G</mi><mo stretchy="false">^</mo></mover><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.
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