Mathematical Model Formulation for Sterilisation Processes in Fermenters

Problem Scenario

You are a research and development engineer for a fermenter supply company.  This company specialises in development of automatically controlled fermenters for various food products.  Your company makes the marketing claim that “fermentations using our equipment rarely fail due to contamination”.  Your manager, the Director of Engineering, Mrs Diana Lang has asked you to investigate the fermentation medium sterilisation procedures, as these are critical to the successful accomplishment of the claimed performance

The standard fermenter design (Fermenter 1) consists of a stirred vessel with a heating coil inserted in the vessel.  In situ sterilisation is accomplished by filling the vessel(s) with fermentation medium, applying heat in the form of a hot fluid flowing through the heating coil, and heating for a period of time.  Once a satisfactory heat treatment is obtained, the hot fluid in the heating coil is replaced by cold water to cool the vessel and medium to the desired fermentation temperature.

A client has requested the addition of a second low cost fermenter (Fermenter 2). Fermenter 2 is not stirred and does not contain a heating coil. In orderto heat treat the fermentation medium in Fermenter 2, it is connected in recycle with Fermenter 1 as shown in the diagram. Thus, the medium in both fermenters can be treated at the same time, and, once cooled, the fermenters can be de?coupled and run as two separate batch fermenters

It is necessary to be able to predict the length of the heat treatment necessary for the combined system so that the sterilisation process can be optimised.   Mrs Lang is particularly concerned about one very expensive medium that can only be pasteurised using hot water at up to 70° C as the heating medium.   This medium is heat sensitive and so even a few minutes of unnecessary treatment at high temperature will damage it significantly. In this medium, spores cannot be killed, but vegetative cells can be. It has been decided to include the destruction of Listeria monocytogenes (D65  = 16.5 s, z = 10°C) in the model.

A mathematical model is required to predict the temperature and viable Listeria concentration changing in each fermenter as a function of time for the heating stage of the process.

Figure One – Schematic diagram of series arrangementfor heat treating fermentation media. (note M = mass flow rates and T = Temperatures that will be measured during experimental validation. Two temperatures are measured in each tank to test the assumption of complete mixing. The inlet and outlet temperatures to the coil are measured to allow calculation ofthe U value for the submerged coil from experimental data during validation experiments.)

You are required to formulate models to describe the following processing system. You should submit a clearly presented model development using the modelling process described in the guidelines and demonstrated in lectures. Your document should include:

definition of the purpose of the model

a.description of the system

b.diagram of the systems including the processing variables

c.assumptions made in the model development along with a brief justification of each

d.definitions of all nomenclature used, including units and types of variables

e.summary of how many equations and of what type are needed to complete the model

f.formulation of equations; stating basis of equation, word balances, the units required for each term and the derivation of the final equation

g.initial conditions

h.summary of model equations requiring solution

You are not required to solve the model developed in Part A.

A model formulation will be provided for Part A. Using this formulation, you are required to develop a solution, check the solution for mathematical and numerical errors and to validate it against experimental data. A lot of time can be wasted debugging MATLAB programmes. We will try to do most of this during class time in scheduled computer lab sessions but if you cannot find a bug in your problem and you have tried hard then ask for help.  

You will need to hand in a short report that:

a.outlines the solution method used and why

b.lists the MATLAB code written to solve the model

c.shows an example solution output along with a commentary on the trends given and whether they are expected

d.demonstrates there are no numerical errors in the solution

e.demonstrates that there are no mathematical errors in the solution

f.summarises all model system inputs for the validation experiment along with the source and/or calculation method

g.validates the model by comparing the solution with experimental data using a sensitivity analysis. To allow you to validate the model, experimental data collected for the system will be provided.

h.identifies whether there is a model weakness or not and why

i.explains the next course of action (model reformulation or improving system input parameter estimation (which ones?)) and why

Your assignment should be well structured, logical and guide the reader through the modelling process and the decisions you have made. It must be an individual effort.