Preliminary Heat Exchanger Design

Introduction
This article will help you understand how you can estimate the initial design characteristics of a heat exchanger.  An Excel spreadsheet using the equations developed in this article is also provided. The equations are derived from a simple heat balance, and a few other elementary relationships.

These equations act as initial estimates, and the results will need to be refined by more sophisticated calculations.  If you just want the spreadsheet then click here, but read the rest of the article if you want to understand the theory.

Theory
Consider a heat exchanger operating in countercurrent flow.

Assume that we know the

• desired input and output temperatures of all streams, 
• specific heat capacities, 
• the overall heat transfer coefficient, 
• and the mass flowrate of the cold stream.  

We will now calculate

• the flowrate of the hot stream, 
• the heat transfer rate 
• and the heat transfer area.

A heat balance on the cold and hot streams gives

Qc gives the overall heat transfer rate.

But Qh = Qc.  Equating both equations and rearranging for the mass flowrate of the hot liquid stream gives

The log mean temperature difference is

The overall heat transfer rate Qc can be defined in terms of the log mean temperature difference

The final two equations can be easily rearranged to give the overall heat transfer area A in terms of the heat transfer rate Qc, the heat transfer area A and the log mean temperature difference.

Excel Implementation
Implementing these basic heat transfer equations in Excel is easy, and no special explanation is required.

Download Excel Spreadsheet for Preliminary Heat Exchanger Design

Related article: Modeling the Temperature Dynamics of a Cross-Flow Heat Exchanger

Dynamic Model of a Cross-Flow Heat Exchanger

Introduction
This article will develop a dynamic model of a cross-flow heat exchanger from first principles, and then discretize the governing partial differential equation with finite difference approximations.  It will then demonstrate how this equation can be implemented in Excel (or indeed any other math tool)

If you just want the Excel implementation, then click here, but I encourage you to read the rest of the article so you understand how the spreadsheet is implemented.

First Principles Modeling
Consider liquid flowing (at mass flowrate F) through a length Δx of pipe (diameter D), subject to cooling by cross-flow air (at temperature Ta and heat transfer coefficient U)

A heat balance over time Δt gives the following.

 

Dividing by Δx and Δt and simplifying gives

As Δx and Δt tend to zero, we get the following parabolic partial differential equation

Equation 1

Finite Difference Approximation

A forward difference approximation for the first of temperature with respect to time is

Equation 2

A backward difference approximation for the first of temperature with respect to space is

Equation 4

Substituting Equation 2 and 3 into Equation 1, and rearranging gives

Equation 4

We only need to know the temperature of the bar at time t (on the RHS of Equation 4) to calculate the temperature at time t + Δt (on the LHS of Equation 4).

Implementating in Excel

This is how Equation 4 will be implemented in Excel

We will now discuss the individual steps in detail.

Step 1  - Specify your parameters, including your chosen time and space step.  I've named the cells in Column C with the names in Column E.  I'll use named values when entering Equation 4.

Step 2 - Create a column and row containing your space and time steps

Step 3 - Fill in your initial conditions at time t = 0 (this will be the inlet liquid temperature as specified in the parameters).

Step 4 - Insert your boundary conditions at distance x = 0 (this will be the inlet liquid temperature - the same as the initial condition).

Step 5 - Implement Equation 4 into the first empty cell (at t = Δt and x = Δx)

Step 5 - Copy this formula to all other times and positions.  For my implementation, I go up to t = 1 and x = 0.4.

The techniques I've demonstrated above can be applied to many other challenges in science, engineering and math.  If you have any requests, then let me know.

Download the Excel 97 spreadsheet that models a Cross-Flow Heat Exchanger