# 47 Lang Factor and Return on Investment

Learning Objectives

By the end of this section, you should be able to:

**Estimate** the total capital investment (TCI) and fixed capital investment(FCI) based on the lang factor

## Lang Factor

Lang factor relates purchased equipment cost to total capital investment (TCI) and fixed capital investment(FCI). This factor represents the cost of building a major expansion onto an existing chemical plant, including other costs such as piping, control, etc.

[latex]C_{TM}=F_{lang}\sum_{i=1}^nC_{p,i}[/latex]

where

[latex]C_{TM}[/latex] – total plant cost

[latex]F_{lang}[/latex] – lang factor

[latex]\sum_{i=1}^nC_{p,i}[/latex] – sum of the cost of all purchased equipment

Lang factors for common plant types are given below. Note that fluid processing means plants processing liquids and/or gasses.

**Table 1: Lang Factors for Common Plant Types [latex]^{[1]}[/latex]**

type of plant | FCI | TCI |
---|---|---|

solid processing plant | 3.9 | 4.6 |

solid-fluid processing plant | 4.1 | 4.9 |

fluid processing plant | 4.8 | 5.7 |

Exercise: Estimating FCI Using the Lang Factor

Suppose we want to build a sulfuric acid production plant. Say the reaction mechanism for the process is as follows:

\begin{align*}

&S_{(s)} + O_{2(g)} → SO_{2(g)}\\

&SO_{2(g)} + \frac{1}{2}O_{2(g)} ⇌ SO_{3(g)}\\

&SO_{3(g)} + H_{2}SO_{4(l)} → H_{2}S_{2}O_{7(l)}\\

&H_{2}S_{2}O_{7(l)} +H_{2}O_{(l)} → 2H_{2}SO_{4(l)}

\end{align*}

The purchased price for all equipment is shown below:

Equipment categories | Puchased price |

pre-treatment | $3,000,000 |

reactors | $8,000,000 |

absorption tower (for separation) | $5,000,000 |

tanks | $2,000,000 |

pumps and compressors | $4,000,000 |

Estimate the total capital investment for the plant.

### Solution

From the reaction mechanisms, we see solid, liquid, and gasous phases in the process streams, so we select the lang factor to calculate TCI for solid-fluid processing plant ([latex]F_{lang}=4.9[/latex]).

\begin{align*}

TCI &=F_{lang}\sum_{i=1}^nC_{p,i}\\

TCI &=4.9×($3,000,000+$8,000,000+$5,000,000\\

&\;\;\;\;\;+$2,000,000+$4,000,000)\\

TCI & = $107,800,000

\end{align*}

## Return on Investment (ROI)

Return on investment is the ratio of the net profit and total capital investment ([latex]TCI[/latex]). The net profit can be calculated by the total revenue minus cost of manufacturing ([latex]COM[/latex]). It provides a measure of how favourable the investment is.

Typically [latex]ROI[/latex] ranges from 10%-20%. An [latex]ROI[/latex] over 30% for a plant is considered very desirable.

[latex]ROI=\frac{(revenue-COM)}{TCI}×100\text{%}[/latex]

Revenue is the total profit we generated from selling the product without deducting the cost of production:

[latex]\text{revenue}=\text{unit price of product} × \text{sales}[/latex]

[latex]COM[/latex] is the total cost of operating the plant, including raw materials, utilities, labours, etc. We are going to learn about how to estimate [latex]COM[/latex] later.

Exercise: Estimating ROI

Suppose we want to build a solid-fluid processing plant that produces soda ash ([latex]NaCO_{3}[/latex]), and it also produces [latex]HCl[/latex] and [latex](NH)_{2}SO_{4}[/latex] as profitable by-products. The annual production rates and market prices for our products are shown below:

product | production rate (tonne/year) | price per tonne |

[latex]NaCO_{3}[/latex] | 120,000 | $220 |

[latex]HCl[/latex] | 95,000 | $145 |

[latex](NH_{4})_{2}SO_{4}[/latex] | 90,000 | $150 |

Suppose the cost for the total purchased equipment is 18,000,000 dollars and the annual cost of manufacturing is 60,000,000 (which includes the cost of raw materials). Assuming all the products are sold, what is the return on investment for our plant? Based on this estimate would it be economically feasible?

### Solution

**Step 1:** Calculate the total revenue generated by all types of products:

\begin{align*}

\text{revenue} &=\sum_{i} \text{unit price of product } i × \text{sales}\\

&=120,000×$220+95,000×$145+90,000×$150\\

& = $53,675,000

\end{align*}

**Step 2:** Estimate [latex]TCI[/latex] using the lang factor:

We select the lang factor to calculate TCI for solid-fluid processing plants ( [latex]F_{lang}=4.9[/latex] ) as stated in the question.

[latex]TCI=4.9×$18,000,000=$88,200,000[/latex]

**Step 3:** Calculate [latex]ROI[/latex] using the values we calculated in the previous steps:

\begin{align*}

ROI &=\frac{(revenue-COM)}{TCI}×100\text{%}\\

&=\frac{($53,675,000-$60,000,000)}{$88,200,000}×100\text{%}\\

& = -7.17 \text{%}

\end{align*}

The project is not feasible because the return on investment is negative, which means we are deficient even if we assume we sold all of the products.

## Other Useful Resources for Capital Cost Estimation

Chemical Engineering Magazine; Accessible from UBC library at http://tinyurl.com/ybaf56kt

G. P. Towler and R. K. Sinnott, Chemical Engineering Design: Principles, Practice, and Economics of Plant and Process Design. (2nd ed.) Waltham, MA; Oxford; Butterworth-Heinemann, 2013; Accessible from UBC library at http://tinyurl.com/y9oubo7n

## References

[1] Joseph A. Shaeiwitz; Debangsu Bhattacharyya; Wallace B. Whiting; Richard C. Bailie; Richard Turton. *Analysis, Synthesis and Design of Chemical Processes*, fifth edition. [online]<https://gw2jh3xr2c.search.serialssolutions.com/?sid=sersol&SS_jc=TC0002267093&title=Analysis%2C%20synthesis%20and%20design%20of%20chemical%20processes> [Accessed 11 June, 2020].

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