## Modeling a Double Pipe Heat Exchanger with MapleSim

A prospective customer recently asked if we had a MapleSim model of a double pipe heat exchanger. Heat exchangers are a critical unit operation in the process industries, and accurate models are needed for process control studies.  I couldn't find an appropriate model so I decided to derive the dynamic equations, and implement them using MapleSim's custom component interface.  I'll outline my modeling strategy in this blog post.

Typically, double pipe heat exchangers are modeled using

• a continuum approach, in which the temperature variation across both streams is described by PDEs,
• or a discretized approach, in which the temperature variation is described by ODEs.

Given that MapleSim solves ODEs (and not PDEs), I chose the discretized approach.

1 Deriving the System Equations

1.1 Introduction

The heat exchanger was divided into N control volumes.

A heat balance on a typical control volume resulted in three differential equations – one each for the tube- and shell-side liquid, and one to model the heat capacity of the tube wall.  Axial heat flow along the tube wall (heat flow into and out of tube wall sections due to temperature differences in adjacent tube wall sections) was also modeled with Fourier’s Law of conduction.

For the simple model outlined here, I’ve assumed that heat exchanger was insulated, so no convective heat losses from the surface are considered.

1.2 Energy Balance on the Tube and Shell-Side Streams

For a single control volume, a heat balance on the tube-side stream gives

However Ttin,i and Ttout,i (the inlet and outlet tube-side temperatures in each control volume) are not state variables. They are approximated by taking the average of adjacent temperatures:

Hence the heat balance on the tube-side becomes

A similar heat balance on the shell-side fluid gives

1.3 Energy Balance on the Tube Wall

The tube wall acts as a heat capacitor, and can have a significant effect on the transfer of energy from one stream to another.  It is assumed that the tube wall has a homogenous temperature in each control volume, with heat transferred to and from the tube- and shell-side liquids, and via conduction from adjacent tube-wall sections. A heat balance gives:

In reality, the inner and outer surface would have different temperatures. This could be modeled by dividing the tube wall into several layers, performing a heat balance on each with Fourier’s Law governing heat flux between layers. This simplified model ignores this effect.

1.4 Heat Transfer Coefficients

The heat transfer coefficients hwt and hws were predicted by the Dittus-Boelter correlation.

2 Implementation of System Equations in a Custom Component

The entire set of differential equations for all N control volumes were generated in a MapleSim custom component with a simple application of the ?seq command (this approach meant I could explore how increasing the number of control volumes affected the results simply by changing the value of N and regenerating the custom component).

Figure 3 give a small subset of equations implemented in the custom component (see the attached MapleSim model for the full set), while Figure 4 outlines the steps involved in creating a custom component.

It’s important to note that no causality has been specified, so the block can be used in any configuration.

3 Complete Heat Exchanger Model

A more sophisticated version of this model can be found here.  This version accounts for the temperature variation of the tube-side liquid viscosity (which can have a significant effect on the tube-side heat transfer coefficient, as predicted by the Dittus-Boelter correlation), and also implements a temperature control loop.

Another version (that I can’t share) models the temperature variation of the tube wall to a greater fidelity, and convective losses from the shell surface.

4 Notation

 Dit Inside diameter of tube m Dot Outside diameter of tube m Dis Inside diameter of shell m t Density of tube-side fluid kg m-3 s Density of shell-side fluid kg m-3 w Density of tube-wall material kg m-3 Cpt Specific heat capacity of tube-side liquid J kg-1 K-1 Cps Specific heat capacity of shell-side liquid J kg-1 K-1 Cpw Specific heat capacity of tube-wall material J kg-1 K-1 hsw Heat transfer coefficient of shell-side fluid and tube wall W m-2 K-1 htw Heat transfer coefficient of tube-side fluid and tube wall W m-2 K-1 L Length of heat exchanger M N Number of control volumes Tti Temperature of tube-side liquid in control volume i K Tsi Temperature of shell-side liquid in control volume i K Twi Temperature of tube wall in control volume i K Qt Flowrate of tube-side liquid m3s-1 Qs Flowrate of tube-side liquid m3s-1

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