# Shell & tube heat exchanger equations

In this article, we will take a details look at equations required for shell and tube heat exchanger sizing and design.

Shell and tube heat exchangers are widely used and very popular in the process industry, due to their versatility. Different types of shell and tube exchangers can be easily configured by changing the shell and tube arrangement.

## Shell & tube heat exchanger design procedure

Shell and tube heat exchanger design is an iterative process, which goes through the following steps.

1. Define process requirements for the new exchanger
2. Select a suitable type of shell and tube exchanger
3. Define design parameters such as - number of tube passes, tube size, shell ID etc.
4. Heat exchanger calculations and modeling to get the output - outlet hot/cold fluid temperature, heat transfer rate, pressure drop on shell/tube sides etc.
5. Check of the output is in accordance with the process requirements
• If the output is as per process requirements and cost is within budget then finalize the process design and prepare a heat exchanger specification sheet
• If the design does not match with either the process requirement or if it is over budget then go back to step 3, change the design parameters and repeat this process again.

There are a few equations that are very important for the calculations that we need to perform during heat exchanger design process.

Here is a list of all the important shell and tube heat exchanger equations.

## Overall heat transfer equation

Overall heat transfer in any exchanger is governed by the following equation -

Equation-1

where, Q = overall heat transfer rate
U = Overall heat transfer coefficient
AOverall = Overall heat transfer surface ares
LMTD = Logarithmic Mean Temperature Difference

## LMTD equation

The logarithmic mean temperature difference is an average quantification of the temperature difference between the shell and tube sides. It is calculated with the following equation.

Equation-2

Where,
ΔT1 → the temperature difference between hot and cold fluids at one end of the heat exchanger
ΔT2 → the temperature difference between hot and cold fluids at the other end of the heat exchanger.

## LMTD with the Correction factor

However the LMTD is valid only for heat exchanger with one shell pass and one tube pass. For multiple number of shell and tube passes the flow pattern in a heat exchanger is neither purely co-current nor purely counter-current. Hence to account for geometric irregularity, Logarithmic Mean Temperature Difference (LMTD) has to be multiplied by a Mean Temperature Difference (MTD) correction factor (FT) to obtain the Corrected Mean Temperature Difference (Corrected MTD).

Equation-3

This correction factor calculator will help you to quickly calculate the LMTD correction factor for a shell and tube exchanger with multiple shell or tube side passes.

## Number of tubes based on the heat transfer area required

The number of tubes needed in shell & tube exchanger (NT) can be calculated using the following equation, based on overall heat transfer area requirement.

Equation-4

Where, we get the AOverall (overall heat transfer area required) from the heat transfer rate equation (Equation-1).
OD is the outside diameter of selected tube size
L is the total tube length

This equation is quite straight forward based on the geometry of the selected shell and tube heat exchanger.

## Tube side fluid velocity

Tube side velocity is important for estimation of Reynolds number on the tubeside and then for getting the heat transfer coefficient for the tube side fluid. We can use the following equation for tube side velocity.

Equation-5

Where, m = mass flow rate on the tube side
NP = Number of tube passes
NT = Number of tubes
ρ = Tube side fluid density
ID = Tube internal diameter

Further, the Reynold's number for the tube side fluid is calculated as,

Equation-6

Here, μ is the viscosity for tube side fluid

## Overall heat transfer coefficient equation

When we have a handle on the heat transfer area (AOverall) and temperature difference (LMTD), the only remaining unknown in the heat transfer equation (Equation-1) is the overall heat transfer coefficient (U). We can use the following equation to get the overall heat transfer coefficient for a shell & tube exchanger.

Equation-7

Where, ho = Shell side heat transfer coefficient
hi = Tube side heat transfer coefficient
Rdo = shell side dirt factor
Rdi = tube side dirt factor
OD and ID are respectively the outer and internal diameters for the selected tube size
Ao and Ai are outer and inner surface area values for the tubes
kw is the resistance value for the tube wall

Note, this overall heat transfer coefficient is calculated based on the outer tube surface area (Ao). So it must be multiplied by the Ao value for using in the overall heat transfer equation.

## Using these shell & tube heat exchanger equations

We already saw that the design of a shell and tube heat exchanger is an iterative process. Often, engineers prefer to use a heat exchanger design software to create a heat exchanger model. You can then use this model to simulate the heat exchanger performance and to verify if it will meet your process requirements.

However, if you decide to manually perform the heat exchanger design and related calculations, here are some calculators and tutorials that can help you.

Here are some recommended steps to use the heat exchanger design equations -

1. Fix the inlet/outlet temperature values
2. Calculate LMTD
3. Select a shell and tube heat exchanger (TEMA) tube
4. Decide on shell and tube geometry
5. Calculate heat transfer area based on selected geometry (AOverall)
6. Get the overall heat transfer coefficient (U), using a suitable empirical correlation for given fluid - for example, Sieder-Tate equation
7. Calculate the overall heat transfer rate (Q), using Equation-1
8. Check of Q matches with the heat lost/gained via temperature change on the hot and cold side. This is the basic energy balance on shell / tube side fluids.
9. Check the pressure drop on shell and tube sides. Does is match with the allowable pressure drop as per process requirements?
10. If the design is adequate as per process requirements, check the tentative material costs. Are they within budget?
11. If either of the design or budget checks fail, go back to step 4 and repeat the process till you get a satisfactory shell & tube heat exchanger design.