## 1. Introduction

## 2. A Double-Pipe Heat Exchanger as Distributed Parameter System

#### 2.1. Governing PDEs

#### 2.2. Spatially Distributed Irrational Transfer Functions

#### 2.3. Frequency Responses

#### 2.4. Steady-State Responses

## 3. Approximate Model of the Heat Exchanger

#### 3.1. MOL Approximation

- A three-element vector ${\vartheta}_{t,n}\left(t\right)$ of the nth section state variables,$${\vartheta}_{n}\left(t\right)={\left[\begin{array}{ccc}{\vartheta}_{t,n}\left(t\right)& {\vartheta}_{w,n}\left(t\right)& {\vartheta}_{s,n}\left(t\right)\end{array}\right]}^{T},$$
- A two-element vector ${\vartheta}_{o,n}\left(t\right)$ of the nth section output signals,$${\vartheta}_{o,n}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{to,n}\left(t\right)& {\vartheta}_{so,n}\left(t\right)\end{array}\right]}^{T}={\left[\begin{array}{cc}{\vartheta}_{t,n}\left(t\right)& {\vartheta}_{s,n}\left(t\right)\end{array}\right]}^{T},$$$${\vartheta}_{to,N}\left(t\right)={\widehat{\vartheta}}_{to}\left(t\right),\phantom{\rule{1.em}{0ex}}{\vartheta}_{so,1}\left(t\right)={\widehat{\vartheta}}_{so}\left(t\right);$$
- A two-element vector ${\vartheta}_{i,n}\left(t\right)$ of the nth section input signals, which for $n=2,3,...,N-1$ are given by the output signals of the adjacent sections,$${\vartheta}_{i,n}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{ti,n}\left(t\right)& {\vartheta}_{si,n}\left(t\right)\end{array}\right]}^{T}={\left[\begin{array}{cc}{\vartheta}_{to,n-1}\left(t\right)& {\vartheta}_{so,n+1}\left(t\right)\end{array}\right]}^{T},$$$${\vartheta}_{i,1}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{ti,1}\left(t\right)& {\vartheta}_{si,1}\left(t\right)\end{array}\right]}^{T}={\left[\begin{array}{cc}{\vartheta}_{ti}\left(t\right)& {\vartheta}_{so,2}\left(t\right)\end{array}\right]}^{T},$$$${\vartheta}_{i,N}\left(t\right)={\left[\begin{array}{cc}{\vartheta}_{ti,N}\left(t\right)& {\vartheta}_{si,N}\left(t\right)\end{array}\right]}^{T}={\left[\begin{array}{cc}{\vartheta}_{to,N-1}\left(t\right)& {\vartheta}_{si}\left(t\right)\end{array}\right]}^{T}.$$

#### 3.2. Approximate Rational Transfer Functions

#### 3.2.1. Single Section

#### 3.2.2. N-Section Approximation Model

## 4. Results and Discussion

#### 4.1. Frequency Responses

#### 4.2. Steady-State Temperature Profiles

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Double-pipe counter-flow heat exchanger: ${v}_{s},{v}_{t}$—shell-side and tube-side fluid velocities; ${\vartheta}_{s},{\vartheta}_{t}$—shell-side and tube-side fluid temperatures; ${\vartheta}_{w}$—wall temperature; ${\vartheta}_{si},{\vartheta}_{ti}$— shell-side and tube-side fluid inlet temperatures; ${\vartheta}_{so},{\vartheta}_{to}$—shell-side and tube-side fluid outlet temperatures; ${d}_{ti},{d}_{to}$—inner and outer tube diameters; ${d}_{si},{d}_{so}$—inner and outer shell diameters; L—heat exchanger length.

**Figure 2.**Spatially distributed transfer function model for the double-pipe counter-flow heat exchanger.

**Figure 3.**Block diagram of the approximate transfer function model for the double-pipe counter-flow heat exchanger.

**Figure 4.**Frequency response ${g}_{ts}(L,i\omega )$ of the irrational transfer function model of the double-pipe counter-flow heat exchanger vs. frequency responses ${\widehat{g}}_{ts}({l}_{N},i\omega )$ of its approximate rational transfer function models for different values of N.

**Figure 5.**Frequency response ${g}_{ss}(0,i\omega )$ of the irrational transfer function model of the double-pipe counter-flow heat exchanger vs. frequency responses ${\widehat{g}}_{ss}({l}_{1},i\omega )$ of its approximate rational transfer function models for different N.

**Figure 6.**Steady-state responses $\overline{\vartheta}\left(l\right)$ of the irrational transfer function model of the double-pipe counter-flow heat exchanger vs. steady-state responses $\overline{\widehat{\vartheta}}\left({l}_{n}\right)$ of its approximate rational transfer function models for different N. Fluid inlet temperatures: ${\overline{\vartheta}}_{ti}=100{\phantom{\rule{4pt}{0ex}}}^{\circ}\mathrm{C}$, ${\overline{\vartheta}}_{si}=50{\phantom{\rule{4pt}{0ex}}}^{\circ}\mathrm{C}$.

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