Heat Transfer in Single-Use Bioreactors and E. coli Bioprocesses

Heat Transfer in Single-Use Bioreactors and E. coli Bioprocesses

vorgelegt von M. Sc.

Matthias Müller

an der Fakultät III – Prozesswissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften - Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzende: Prof. Dr. Steffi Knorn Gutachter: Prof. Dr. Matthias Kraume Gutachter: Prof. Dr. Dieter Eibl

Gutachter: Prof. Dr. habil. Wolfram Meusel

Tag der wissenschaftlichen Aussprache: 23. August 2021

Danksagung

Mein Dank gilt allen, die diese Arbeit ermöglicht und zu ihrem Gelingen beigetragen haben.

Prof. Dr.-Ing. Matthias Kraume, Prof. Dr.-Ing. habil. Wolfram Meusel und Prof. Dr.-Ing. Dieter Eibl für die Übernahme der Gutachten, der fachlichen Unterstützung und Ihrem Interesse am Thema mit dem Sie diese Arbeit maßgeblich vorangebracht haben.

Der Hochschule Anhalt und den Mitarbeitern der AG Bioverfahrenstechnik und Bioprozesstechnik, sowie dem ganzen Fachbereich Biowissenschaften und Prozesstechnik.

Sartorius für die Möglichkeit das Thema zu bearbeiten und die notwendigen Mittel zur Verfügung zu stellen, speziell Dr. Gerhard Greller, Ute Husemann und dem ganzen Team von (ehem.) Upstream Technology.

Der tatkräftigen Unterstützung der Studenten Ulf Dehio, Gerald Krepinski, Johannes Nathan und Eric Schulze die mit Ihren Bachelor- und Masterarbeiten einen Beitrag zu dieser Arbeit geleistet haben.

Den Doktorandenkollegen für den Austausch und die Diskussionen auch abseits des eigenen Themas.

Der ZHAW für die Möglichkeit einen Teil der Experimente dort durchzuführen.

Der DECHEMA Arbeitsgruppe „Single-Use Technologie in der biopharmazeutischen Produktion“.

Und allen voran meiner Familie und meinen Freunden.

Abstract

Aim of this work was to evaluate stirred, single-use bioreactors by means of their heat transfer capabilities. Although this bioreactor concept is increasingly adapted by the biopharmaceutical industry, this technical aspect is currently underrepresented in the literature. In contrast to conventional, stainless- steel bioreactors, single-use systems provide some inherent properties that possibly affect heat transfer, such as the polymer film layer of the cultivation chamber and a limited heat transfer area. Therefore, commercially available reaction systems should be investigated.

To assess the relevant systems, an adequate experimental procedure was developed. Generally, two types of methods are available: transient experiments, i.e., heating and cooling curves, can directly be applied to jacketed, stirred bioreactors with minimal modification. However, results derived from this approach are more difficult to evaluate, because of the dynamic system behavior. Secondly, steady state methods can be applied, which require an additional heat source. Often electrical heaters are used. In this work, a chemical alternative was established and applied up to a scale of 200 L: the exothermic decay of hydrogen peroxide. Another published work focused on a comprehensive study of thermal properties of stirred single-use bioreactors, where especially transient and steady state experiments were carried out in parallel and set into relation. Finally, a full-system heat transfer model of a stirred 500 L bioreactor was set up, considering all relevant peripheral components required for heat transfer.

Single-use bioreactors are mainly used for mammalian cell cultures. The challenging requirements towards the reaction equipment imposed by microbial cultures can hardly be fulfilled by current systems.

Nevertheless, these applications are industrially relevant and could generally benefit from the technology. One of the main challenges lies in the metabolic heat produced by the microbial cells. To investigate the biological side, a similar experimental approach was applied to E. coli fed-batch processes, but in a 5 L glass bioreactor.

The present work, therefore, represents a contribution to both, the engineering characterization of single- use bioreactors and the related application of calorimetric methods to microbial processes.

Contents

1 Introduction ... 1-1 2 Fundamentals ... 2-3 2.1 Forms of heat transfer ... 2-3 2.1.1 Heat conduction ... 2-3 2.1.2 Convective heat transfer ... 2-4 2.1.3 Conjugate heat transfer through a plane wall ... 2-5 2.2 Overall bioreactor heat balance ... 2-8 2.3 Investigation of heat transfer coefficients in stirred tank bioreactors... 2-11 2.3.1 Dimensionless representation of heat transfer ... 2-11 2.3.2 Process side heat transfer ... 2-12 2.3.3 Jacket side heat transfer ... 2-13 2.3.4 Wilson plot technique ... 2-16 2.4 Biothermodynamics on single-cell level ... 2-19 3 Experimental methods and data assessment ... 3-25 3.1 Experimental tools of assessing heat flows: Calorimeters ... 3-25 3.2 Determination of overall heat transfer coefficients ... 3-27 3.2.1 Transient experiments ... 3-27 3.2.1.1 General considerations ... 3-27 3.2.1.2 Calculating overall heat transfer coefficients from transient experiments ... 3-27 3.2.2 Steady state experiments ... 3-29 3.2.2.1 Electrical heater as heat source ... 3-29 3.2.2.2 Chemical reaction as heat source... 3-30 3.2.2.3 Calculating overall heat transfer coefficients from steady state experiments ... 3-31 3.3 Determination of the metabolic heat flow in bioprocesses ... 3-32 3.3.1 Extended bioreactor heat balance ... 3-32 3.3.2 Baseline approach ... 3-33 3.3.3 Calorimetric modes ... 3-34 3.3.4 Determination of the metabolic heat flow ... 3-35 3.3.5 Controller design for heat compensation calorimetry in fed-batch processes ... 3-36 3.4 Technical implementation of biocalorimetric measurements ... 3-40 3.4.1 Single-use bioreactor for the determination of overall heat transfer coefficients... 3-40 3.4.2 Single-use bioreactor for the assessment of a complete plant model at 500 L ... 3-43

3.4.3 Bioreactor setup for the determination of metabolic heat flow during fermentation ... 3-46 4 Results and Discussion ... 4-48 4.1 Thermal characterization of stirred single-use bioreactors ... 4-48 4.1.1 Overall heat transfer coefficients in STR^® 50 ... 4-49 4.1.2 Steady state measurements with chemical heat source ... 4-53 4.1.3 Comparison of theoretically and experimentally derived heat transfer coefficients ... 4-55 4.1.4 Combined application of the Wilson plot method and dimensionless representation to experimental data ... 4-59 4.1.5 Influence of polymer film layer ... 4-60 4.1.6 Scale dependency ... 4-61 4.2 Case study: Extended plant model of a STR^® 500, 2^nd Gen. ... 4-63 4.2.1 Development of the overall model from subsystems ... 4-63 4.2.2 Simulation procedure of the overall plant model ... 4-66 4.2.3 Results ... 4-67 4.2.3.1 Overall heat transfer coefficient ... 4-67 4.2.3.2 Full system model... 4-68 4.3 Assessment of metabolic heat during fermentations ... 4-70 4.3.1 Comparison of batch fermentations: Capacity vs. compensation mode ... 4-70 4.3.2 Comparison of heat transfer coefficients ... 4-73 4.3.3 Inherent implications of experimentally derived overall heat transfer coefficients ... 4-74 4.3.4 Fed-batch control ... 4-76 5 Summary and Conclusion... 5-79 References ... 5-81 List of Publications ... 5-86 Appendix-1 Temperature and hydrodynamic effect under steady-state conditions ... 5-88 Appendix-2 Publication reprints... 5-92

Nomenclature Symbols

A area m²

a thermal diffusity m² sec^-1

C geometric parameter -

ci concentration g L^-1

cp specific heat capacity J kg^-1 K^-1

D dilution rate h^-1

d diameter m

e control deviation

F flow rate L sec^-1

F’ free enthalpy J

G Gibbs free enthalpy J

H Enthalpy J

h height m

h̅ partial molar enthalpy J mole^-1

KI integral gain KR proportional gain

l characteristic length m

m mass kg

ṁ mass flow kg sec^-1

N stirring frequency min^-1

n count

ṅ molar rate mole sec^-1

P power W

p pressure Pa

pO2 dissolved oxygen concentration mole L^-1/ %

Q heat J

q heat flow W

q" heat flux W m^-2

Q² coefficient of predictive correlation R² coefficient of correlation

S entropy J K^-1

Ṡ rate of entropy production J K^-1 sec^-1

s wall thickness, gap width, layer length m

s̅ partial molar entropy J K^-1 C^-1 mole^-1

T (absolute) temperature K/ °C

t time sec

U overall heat transfer coefficient W m^-2 K^-1

U' inner energy J

u specific inner energy J kg^-1

V volume L Vi Viscosity-ratio

v specific volume m³ kg^-1

W work W

w velocity m sec^-1

x distance m

y wall distance m

Y yield coefficient

y+ dimensionless wall distance

Yi/j yield of i^th metabolite per j^th metabolite, or stoichiometric coefficient Z number of impeller blades

z distance m

ΔkHO2 oxycaloric equivalent kJ mole^-1

ΔrHx standard enthalpy change of growth reaction J mole^-1

ΔrHx0 standard enthalpy change of growth reaction J mole^-1

ΔrGx Gibbs energy of growth reaction J mole^-1

ΔrGx0 standard Gibbs energy of growth reaction J mole^-1

ΔT temperature difference K

ΔTln mean logarithmic temperature difference K

Greek

α heat transfer coefficient W m^-2 K^-1

γ impeller blade angle (AoA) °

Δ difference

δ thickness of laminar sublayer m

ε specific power input W L^-1

ζ discharge coefficient

ζ̇ rate of advancement of growth reaction C-mol sec^-1 per cell

η dynamic viscosity Pa sec

λ thermal conductivity W m^-1 K^-1

μ specific growth rate h^-1

µi chemical potential of i^th metabolite J C-mole^-1

ν kinematic viscosity m² sec^-1

νi stoichiometric coefficient of i^th metabolite in a reaction

ρ density kg m^-3

Dimensionless numbers Nu =α l

λ ^{Nusselt -}

w l

Subscripts

0 clearance stirrer to bottom, initial 0... 6 index of local temperature

∞ vicinity a ambient bio biological

bl baseline cw cold water dcw dry cell weight

e environment el electrical eq equivalent eva evaporation feed feed

h heater ht heat transfer hyd hydraulic

i inlet, inner circuit j jacket

l liquid

n n-th component o outlet, outer circuit out outlet

p heat source

PHE plate heat exchanger r process side s stirrer set setpoint source heat source

v vessel w wall x biomass Δ metabolic heat

Introduction

1 Introduction

Bioreactors perform multiple tasks to provide optimal growth conditions for cells in a controlled manner.

Ensuring a constant temperature is of major concern because it determines the activity of the overall metabolisms, triggers promotors, or leads to thermal damage, if too high.

Despite the importance, characterizing bioreactors by means of heat transfer is often not in the focus of biochemical engineers, since other aspects, such as oxygen mass transfer or mixing are considered of higher priority, especially when it comes to the scale up of bioprocesses. Nevertheless, there are various situations where heat transfer becomes limiting. In case of jacketed stirred tanks, large vessels must be equipped with additional internal cooling coils, because the volume specific heat transfer area decreases with scale. However, such installations impose other problems, for example when it comes to cleaning.

Further, bioprocess take place at mild temperatures. The cooling capabilities are proportional to the temperature difference between the process and cooling fluid. In biotechnology the driving force is small, compared to chemical reactions, where the temperatures of some hundred degrees are not unusual.

A variety of investigations have been carried out by researches, determining the influence of vessel shapes, stirrer and baffle configurations, installations and fluid properties on heat transfer. However, still there are no comprehensive, general models and most of the available relationships are only valid for specific cases under which they were determined. Even though heat transfer has been object of research for decades, there are still white spots. For example, the local nature of heat transfer, happening at fluid-solid boundary, is particularly challenging to resolve experimentally. The emerging field of numerical fluid dynamics may help understand this aspect in the future, but the computational effort is still high, especially when considering technical devices as a whole.

This thesis aims to identify feasible methods to practically assess bioreactor equipment by means of heat transfer. In this context, a specific class of bioreactors is of concern: single-use bioreactors. Being broadly established in the biopharmaceutical industry for cell culture applications, their heat transfer capabilities have yet been insufficiently described. This becomes even more relevant if this type of bioreactors is supposed to be used for thermally challenging applications like microbial fermentation.

The conceptual design of this thesis is outlined in figure 1-1. It represents the workflow from establishing the experimental methods and applying them to practically relevant cases.

Introduction

Figure 1-1. Outline of the Thesis.

Müller, M. (2019). Heat Transfer in Bioreactors.In M. Moo-Young (Ed.), Comprehensive Biotechnology (3rd ed., pp. 133–150). Elsevier B.V. https://doi.org/10.1016/B978-0-444-64046- 8.00140-3

Initial scope Thesis

–

Müller, M., Meusel, W., Husemann, U., Greller, G., & Kraume, M. (2018). Application of heat compensation calorimetry to an E. coli fed-batch process. Journal of Biotechnology, 266(July 2017), 133–143. https://doi.org/10.1016/j.jbiotec.2017.12.002

Application Conventionalmethods –

Methoddevelopment

Fundamentals

2 Fundamentals

2.1 Forms of heat transfer

Heat transfer is the process of energy exchange between systems through their interconnected boundaries. The driving force of this process is only a temperature difference. Thus, heat transfer always reflects an equilibration process by means of temperature. When setting up an energy balance over a system, further forms of energy transport, such as work, need to be considered as well. How to balance a bioreactor will be covered exemplarily in section 2.2. To quantify the transferred heat, the heat flow, q, is used. As per definition, this parameter depicts the amount of heat transferred per time unit.

q =dQ dt

2-1

Alternatively, the heat flux, q”, is often used to take the heat exchange area into account.

q" =q

A ^2-2

To describe this process in a mathematical manner, different mechanisms of heat transfer need to be distinguished:

• heat conduction within solids or fluids

• forced or free heat convection between fluids and solids

• radiation

As the latter only plays a role in high temperature applications, which are seldom encountered in biotechnology, radiation will not be part of this chapter.

2.1.1 Heat conduction

Heat conduction represents the transfer of kinetic energy between molecules of a material that holds a temperature gradient. The general mathematical relation for this phenomenon is given by Fourier’s law:

q"

⃗⃗⃗ = −λ ∇⃗⃗ T ^2-3

A material thereby is characterized by the thermal conductivity, λ. Although the law is universally valid, for technical calculations the one-dimensional heat conduction under stationary conditions is of major importance, leading to the simplified, integrated formulation,

q = λ s⁄ ΔT ^2-4

Fundamentals

2.1.2 Convective heat transfer

In contrast to heat conduction, energy transport by convective heat transfer goes along with an actual material flow. This flow can be introduced in two ways. First, free convection happens due to density differences that finally result from a temperature gradient, creating internal buoyancy. In bioreactors, this might happen on the outside of a vessel, which usually corresponds to some kind of heat loss to the environment. Further, some jacket designs (open jackets) could be prone to internal free convection, if very low flow rates are present. However, in technical environments, such as jacketed, stirred tank bioreactors, forced convection is the predominant form, where external devices, e.g., stirrers or pumps, ensure the flow. For practical considerations, the energy exchange by convection within a fluid is important to diminish temperature gradients, but usually the heat transfer between a solid and a fluid is the main objective. This process is basically affected by both, fluid properties and process conditions, especially in the near wall region. In analogy to the conductive heat transfer, Newton’s law of cooling describes a linear relationship between a heat flow and its corresponding driving force which is defined as the difference between the temperature at the wall, Tw, and the vicinity, T∞, multiplied by a characteristic parameter called the heat transfer coefficient, α.

q = α ∙ (T_w− T_∞) ^2-5

This linear equation is valid per definition and does not actually represent a law of nature, as the parameters may not be fully independent from each other. As an example, the relationship between heat flow and temperature difference becomes non-linear when phase change occurs (boiling or condensation). Partially because latent heat is exchanged between the phase changing fluid and the wall without change in the fluid’s temperature [1]. The complex relationship between heat flow and temperature difference is often referred to as Nukiyama-, or boiling curves [2]. Nevertheless, the linear relationship is well applicable to single phase, forced convection processes. As was found by Prandtl, turbulent flows are characterized by a laminar sublayer at the wall surface. In the normal direction of a laminar flow, heat transfer takes place in form of conduction only. Hence, if the thickness of the sublayer, δ, is known, the laws of heat transfer by conductivity could be applied, by replacing the heat transfer coefficient term as follows:

α =λ

δ ^2-6

However, this is the very problem since the thickness arises from complex interactions between the fluid and flow properties. The schematic representation of the heat transfer from a wall to the fluid through the boundary is given by figure 2-1. A tangent, simplifying the complex profile to a linear relation, approximates the temperature profile. This model further shows that the actual temperature gradient only is present within the laminar layer.

Fundamentals

Figure 2-1. Temperature profile in the near wall region, where heat is transferred from a hot wall to a turbulent flowing fluid.

A linear relation approximates the complex profile of the laminar sublayer since its actual thickness is hardly accessible.

When a fluid enters a system, a certain onset is required until both, laminar and thermal sublayers are fully established. If the onset region is short compared to the overall length of the flow path (d/l >> 0.1) it is often neglected. However, equations are available that approximate these effects with reasonable deviation and have been applied in the corresponding section 4.1.3.

2.1.3 Conjugate heat transfer through a plane wall

So far, only the different mechanisms of heat transfer were discussed. However, in bioreactors, these individual processes happen simultaneously. Therefore, they need to be combined to an overall heat transfer process. The general scheme for the simplest case of heat from fluid to fluid through a wall is given by figure 2.2.

q" = α₁∙ (T_∞1− T_w1) =λ_w s_w

⁄ ∙ (T_w1− T_w2) = α₂∙ (T_w2− T_∞2) ^2-7

with

A = {A₁; A_w; A₂}

w al l

T

x

T

T

λ⁄ = δα

Fundamentals

Figure 2-2. Heat transfer between two fluids through a plane wall driven by a temperature gradient.

This definition of constant heat exchange area for the sub processes approximates a geometry by a flat plane. This is usually valid for technical scale reactors, where the overall diameter is large compared to the wall thickness and thus, the curvature is small. For smaller glass vessels logarithmic averages may be used. For technical applications, it is therefore important to clearly specify how the heat exchange area was defined. Rewriting the above equation by replacing the in-between wall temperatures results in the following generalized equation, considering heat conduction through multiple layers:

q" = 1 1

α₁+ ∑ s_i λi 𝑛𝑖=1 + 1

α₂

∙ (T_∞1− T_∞2) _2-8

In equation 2-8, only the bulk temperatures on both fluid sides, T∞1 and T∞2, need to be considered making it unnecessary to evaluate the wall temperatures. Thus, the total thermal resistance is summarized by the overall heat transfer coefficient, U:

1 U= 1

α₁+ ∑s_i λ_i

𝑛

𝑖=1

+ 1

α₂ ^2-9

In case of stainless-steel bioreactors, heat conduction through the vessel wall can often be neglected since the heat conductivity of steel is about 15 W m^-1 K^-1 and a single steel wall is usually only a few millimeters thick. Both aspects combined result in a very low thermal resistance. However, if scaling on the wall is present, even thin layers can add a significant thermal resistance, because of very low heat conductivities. Therefore, an assessment should be carried out. Scaling can accumulate over time and

T

x

T

T

T

T

‘‘

Fundamentals thus, the thermal behavior of reaction equipment might change during operation. Another important conclusion arising from the mathematical structure of the above equation is that the overall heat transfer coefficient is always smaller than the smallest individual term, i.e., α1, sw/λw or α2, respectively. This is important when optimizing heat transfer processes, since improving heat transfer on the non-limiting side would have less impact. An extreme example is heating a highly viscous process fluid by condensing steam in the jacket, where only modification on the process side, e.g., by introducing scraping elements on the stirrer, would significantly enhance heat transfer.

Fundamentals

2.2 Overall bioreactor heat balance

There are various reasons for setting up overall heat balances practically. For example, if a cooling system must be designed it is important to estimate its requirements and building the cooling circuit accordingly. Next to this constructive aspect, some heat flow terms can give valuable insight into the process. If the heat of reaction is determined, the turnover or the general performance of a reaction can be estimated. However, this only relates to exothermic/ endothermic reactions, where measurable amounts of heats are released/ consumed. This field of research is called reaction calorimetry. While there are laboratory scale, high precision calorimeters, (bio-) reactors that are equipped with additional measurement capabilities are sometimes referred to as mega-calorimeters. Mostly, such measurements are carried out non-invasive and, additionally, deliver online information. This makes it very attractive in biotechnology, where sterility is important and additional insight into the metabolism of a culture offers sophisticated control possibilities. In the following, some of the most important heat flow terms present in bioreactors are discussed as proposed by Voisard et al. [3].

Setting up a heat balance over any system means accounting for all entering, exiting and accumulating heat flows. The most fundamental representation is given by equation 2-10:

dH

dt = ∑ q_i

𝑛

𝑖=1

2-10

Common bioreactor design incorporates two balance spaces: the process and the jacket side. Both share a common heat flow term. A schematic representation of a batch reactor is given by figure 2-3.

Figure 2-3. Schematic representation of heat flows occurring at a jacketed, stirred tank reactor.

q q

q

q

q

q

q

Fundamentals Thus, two equations can be derived for the reaction side, r, (equation 2-11), and the jacket side, j, (equation 2-12):

dH

dt = −q + q_p+ q − q ^2-11

dH

dt = q + q ₁− q ₂− q ^2-12

With,

H_i= ∑(m_i∙ c_{p i}∙ T_i)

n

i=1

2-13

The accumulation term is represented by the sum of the heat capacities, as product of mass, m, specific heat capacity, cp, and corresponding temperature, Ti, of all temperature-varying parts that are assigned to this balance space, e.g., installations, vessel wall and liquid. Small vessels are more impacted by this, since the heat capacities of these parts have a higher share compared to the fluid, than in larger vessels.

For simplification, it is assumed all parts within the balance space are in equilibrium by means of a common temperature. If so, a weight averaged heat capacity can be used.

m_{i a} ∙ c̅_{p i}≈ ∑(m_i∙ c_{p i})

i=1

2-14

This approach further requires the definition of average reference temperatures for both balance spaces, T̅ or T̅.¹ For low-viscosity process fluids and assumed ideal mixing in the vessel, the process side temperature can be measured at any point in the vessel (equation 2-15). For the jacket side, various scenarios are possible, because it is a flow-through system. Considering a significant temperature difference between inlet and outlet, a logarithmic or arithmetic mean temperature between inlet and outlet may be calculated (equation 2-16). On the other hand, if the jacket is operated at high flow rates, ideal mixing can be assumed where the average jacket side temperature would be represented simply by the jacket outlet temperature. Both approaches might be feasible, e.g., to define a reference for the temperature depended fluid properties.

T̅ = T ^2-15

T̅ =^{Tj2−Tj 1}

n(^{Tj 2}

Tj 1)

or ^{Tj 2+Tj 1}

2 with T2 > T1 2-16

Biological processes are mostly carried out at a defined temperature, i.e. under steady state, isothermal

Fundamentals While the cells grow during a process the release of metabolic heat gradually increases. To compensate this, active cooling is performed by decreasing the jacket side temperature accordingly. Nevertheless, if the change is slow, the accumulation term of the jacket is neglected.

m ∙ c_p dT

dt ≈ 0 ^2-18

Finally, the steady state heat balances according to figure 2-3 are as follows:

Process side

0 = q_p+ q − q − q ^2-19

Jacket side

0 = q + q ₁− q ₂− q ^2-20

For aerobic, microbial fermentations, the reaction specific heat flow, qp, can reach very high values, up to around 50 W L^-1. This value might even be surpassed for very high cell densities and growth rates.

Determining qp often is object of scientific research, including this work, as it gives insights into the metabolic state of the cell of interest. Section 3.3 discusses the details about how to setup a bioreactor balance that allows for biocalorimetry.

The power input by the stirrer, qs, is another significant heat source. Often, varying the stirring frequency is part of the oxygen control cascade which makes online monitoring of this parameter necessary. This can be done by either torque measurement or correlating power input via the consumption of electrical energy that is drawn from the motor. However, accurate determination of the empty load must be carried out prior to a run since bearings and other friction losses would negatively impair the results. Depending on the construction of the jacket, if it fully covers the vessel, heat loss, q , can be neglected for the process side. The heat transfer term, qrj, connects both balance spaces and is represented by equation 2-21².

q = U ∙ A ∙ ΔT ^2-21

To assess this heat flow directly, the overall heat transfer coefficient, Urj, must be known, which is often not the case. The value is affected by the fluid and process conditions, which might change during a process. Further, for jacketed vessels the heat transfer area, Arj, increases due to addition of feeding media or corrective liquids. Proper experimental determination of Urj is one major objective of this work and will be discussed in more detail later (e.g., section 3.2). The driving force, ΔT , must be defined according to the flow characteristics of the jacket. Commonly, the logarithmic mean temperature difference is applied (cf., equation 2-38).

Fundamentals Evaluating the jacket side requires knowledge of inlet and outlet temperatures, Tj,1 and Tj,2, respectively, and the mass flow of the heat carrier media (equation 2-22).

q ₁− q ₂= ṁ ∙ c_p ∙ (T ₁− T ₂) ^2-22

Heat loss from the jacket to the environment, qj,loss, is described analogue to the wall´s heat transfer term, but with lower heat transfer coefficients of around Uj,loss ≈ 5... 15 W m^-2 K^-1. If insulated well, or the difference between jacket and ambient temperature is low, this term might be neglected.

Although the above terms represent the major heat flows of a jacketed bioreactor, considering smaller heat flows could be needed for high accuracy applications. Exemplarily, such additional heat flows could be due to the following:

• evaporation

• neutralization enthalpy of corrective media

• pCO2 stripping

• substrate feeding of different temperature than the process fluid

2.3 Investigation of heat transfer coefficients in stirred tank bioreactors

2.3.1 Dimensionless representation of heat transfer

Nusselt first developed the laws of similarity of heat transfer processes from differential equations already in 1910 [4] by comparing the temperature profiles in pipes of different diameter, fluid species and velocity under steady state conditions in a theoretical manner. The set of differential equations comprised of the Navier-Stokes, continuity and energy equation. By introducing proportional factors of characteristic parameters, such as velocities, heat transfer coefficients or fluid density, it was possible to derive three dimensionless relations that describe the process:

Re =w l

ν ^2-23

Pe =w l a

2-24

Nu =α l λ

2-25

Thus, two processes are similar by means of heat transfer, if they are characterized by same values of either two of Re, Pe and Nu. To retrace the full approach of how the dimensionless numbers are derived, it shall be referred to the comprehensive work of Gröber et al [5]. The Nu number in this case can be interpreted as the relation of the actual heat transfer compared to heat transfer through the laminar

Fundamentals With other words, Nu > 1 means an increased heat transfer compared to the same resting fluid. Further, instead of Pe, the Pr number is usually used in forced convection:

Pr =Pe

Re=η ∙ c_p

λ ^2-27

It shall be noted that this result is derived without actually solving the equations, but by a theoretical evaluation of the underlying physical balances and application of proportionalities. The above approach differs from dimensional analysis (i.e., Π theorem), which might lead to similar dimensionless numbers, but does not require initial knowledge of the relevant differential equations.

A functional relation between these dimensionless numbers can be used to describe the process of heat transfer, although the structure of this relation, f, is yet to be defined:

Nu =α l

λ = f(Re Pr) ^2-28

A once defined structure is considered valid for similar geometries. Nevertheless, a general shortcoming of reported Nu correlations are partially significant errors, up to 40 % [6]. This fact limits the applicability of such correlations even to slightly different geometries or fluid conditions.

2.3.2 Process side heat transfer

The heat transfer on the inner wall of the stirred tank is affected by various factors. Some of the most important are:

• stirrer type, number and stirring frequency

• baffle configuration

• heat transfer mode, i.e., heating or cooling

• fluid properties

• gassing

The following equation is a representation, frequently found in the literature:

Nu = C Re ^a Pr ^b Vi^{c 2-29}

Fundamentals Although the values might differ slightly in newer investigations, values of a = 2/3, b = 1/3 [7], and c = 0.14 [8] still persist. In contrast, the pre-factor, C, accounts for most of the geometric variation, e.g., the stirrer type and baffle configuration. One of the well-known references in this regard is the work by Nagata, covering Rushton-type and pitched blade impellers, containing ratios of geometric dimensions [9]:

Without baffles:

C = 0.54 (d d_v)

−0.25

(h d_v)

0.15

(h₀ h )

0.15

sin(γ)^0.5 Z^{0.15 2-30}

With baffles:

C = 1.4 (d dv

)

−0.3

(h dv

)

0.45

(h₀ h )

0.2

(h dv

)

−0.6

sin(γ)^0.5 Z^{0.2 2-31}

Heat transfer processes are particularly affected by fluid properties which themselves depend on temperature. The Re and Pr numbers only cover temperature effects globally for the bulk fluid. As the main temperature gradient is present within the laminar sublayer, an additional parameter, Vi, is introduced to the equation:

Vi = ( η η_w)

c 2-32

For the exponent, c, a value of 0.14 is commonly found. The term sets the mean bulk viscosity, η, in relation to the viscosity directly at the wall, ηw, by which heating and cooling can be distinguished.

Therefore, heating would support heat transfer by reducing the viscosity in the sublayer and vice versa.

However, the effect is less pronounced during standard operation, except for steam sterilization, and can be considered smaller than ± 5 %. In chemical engineering, where viscosity of polymer solutions can be magnitudes higher compared to water, close-clearance or wall scraping impellers are used under partial or fully laminar conditions, where most of the approaches shown here do not apply directly and therefore, specific models must be used, depending strongly on the given process conditions and fluid properties. Although some process fluids present in biotechnology show highly viscous, non-Newtonian behavior, e.g., some mycelial fungi cultures or during xanthan fermentation, most heat transfer tasks can still be covered by the equations given in this section. As a rule, before using models from the literature, the parameter ranges need to be evaluated upfront, since empiric models are only valid within the ranges they were initially set up.

Fundamentals The simplest design is an open jacket, without any installations. Although this is easy to build, the flow characteristics are usually undesirable as this can lead to large dead volumes and overall low liquid velocities. To guide the flow spirally around the vessel, baffles are introduced into the jacket.

Alternatively, half pipes are welded directly onto the vessel (figure 2-4).

Figure 2-4. Different design principles of jackets. Baffles inside the jacket or half pipes guide the cooling liquid around the vessel in a spiral manner, which increases efficiency.

Despite the variety of construction designs, a certain level of abstraction can already result in an adequate approximation of the underlying heat transfer processes, without setting up sophisticated models. For example, an open jacket can be represented by a concentric annular gap. With defining the hydraulic diameter, dhyd, as characteristic length, the equations for tubular pipes apply as the difference between the outer, do, and inner diameter, di:

d_hyd= d − d_i ^2-33

For fully established turbulent flow in a pipe, Gnielinksi found the following correlation [10]:

Nu _{u bu en} =

(𝜉_gap

⁄ ) Re Pr8 k₁+ 12.7√𝜉_gap

⁄ (Pr8 ^2⁄3 − 1)

[1 + (d_hyd l )

2⁄3

] F_gap ^2-34

with

k₁= 1.07 +900

Re − 0.63

(1 + 10 Pr) ^2-35

F_gap= 0.75 (d_i

⁄ )d

−0.17

2-36

half-pipe baffled

spiral jacket flow open

Fundamentals The flow characteristics need to be carefully calculated upfront. Due to the high volume and undirected flow, high jacket flow rates would be necessary to achieve fully turbulent flow. For this design, transitional flow is to be expected with probably pronounced zones of laminar flow. Because of this, open jackets are unfavorable and are mainly used for small bioreactors, where the introduction of baffles would be too costly. Due to the shape, baffled jackets can be approximated by the flow through a plane gap. If the vessel is not too small, the deviation in flow behavior, i.e. Coriolis-like patterns, can be neglected and a straightforward flow may be assumed. In contrast to the annular gap, the hydraulic diameter for a rectangular cross section is given by the following equation:

d_hyd= 2 s ∙ Δh_{baff e}

s + Δh_{baff e} ^2-37

With the gap width, s, and the riser height between the baffles, Δhbaffle. Again, if fully established turbulent flow is present, the same equation as already given by equation 2-34 apply for this case as well but without the correction term, Fgap. As transitional or even laminar flow is often the dominant characteristic flow pattern, the equations need to be chosen accordingly and can be found in the comprehensive references, especially [11]. However, under laminar flow conditions, the onset distance for both, the laminar flow itself and the thermal boundary layer need to be considered. The procedure results in multiple Nu numbers accounting for these effects that are then later condensed to an average value. This approach was applied practically in this work (section 4.1.3).

Although abstracting the jacket as annular gap or simple pipe flow is often adequate within technical accuracy, dedicated investigations have been carried out to describe the process of heat transfer taking place on the jacket side in more depth. A very fundamental and often cited reference was published by Lehrer [12]. However, the approach is very extensive and thus, cannot be depicted here comprehensively. A major drawback is that only plain, open jackets were object of this investigation.

On the other side, the impact of different jacket inlet implementations, i.e., radial or tangential, was taken into account, as this constructive detail is of major importance for the open jacket design, where significant backmixing occurs.

Another important aspect of the jacket derives from its nature as flow through system. As such, it is characterized by a residence time distribution, which has impact on the definition of the driving temperature difference the jacket and the process side. Ideally, a baffled or half-pipe jacket should show a plug flow with a logarithmic temperature profile from inlet to outlet. Considering the process side as ideally mixed, the common definition of mean logarithmic temperature difference can be derived analytically.

T − T

Fundamentals In contrast, an open jacket with a high flow rate might show more of an ideally mixed behavior. This situation would represent the opposite ideal case. The definition of the driving temperature difference then simplifies to:

ΔT = T ₂− T ^2-39

However, the real residence time distribution would lie between both ideal cases. This becomes obvious when comparing the equations, since the latter results in a lower driving force. Consequently, a jacket that is characterized by a real residence time distribution, using the logarithmic definition would lead to an overestimation of the temperature difference:

ΔT _n> ΔT _ea > ΔT ^2-40

Therefore, some approaches take this effect into account by introducing correcting factors to the driving force [13]. The Pe number is derived experimentally from tracer-based residence time distribution measurements. If only backmixing occurs, this parameter is well suited to characterize a flow through system. However, applying such concepts might only be necessary for certain applications, where the standard definitions fail or are less accurate. Nevertheless, considering the flow pattern of a jacket is beneficial for the understanding of the overall process, and is especially important, if the heat transfer coefficients of both sides, jacket and process side, are of the same magnitude.

2.3.4 Wilson plot technique

Overall heat transfer coefficients generally characterize a conjugate heat transfer process between two fluids through a wall. If such values are determined experimentally it might be desirable to deeper investigate the underlying processes. One example might be the presence of a one-sided limitation, significantly reducing the overall heat transfer capabilities. Due to its nature, the overall heat transfer is limited by its least capable contributor. Therefore, investigations targeting one sided heat transfer coefficients would require the other side to have infinite heat transfer capabilities. This would enable the direct calculation of the desired heat transfer coefficient. As this theoretical concept is only applicable to investigations if dedicated experimental setups are used, alternative approaches were developed enabling a detailed investigation of actual process equipment without extensive need for modification. Although such results might have limited universality, studies in adequate models can simplify further scale-up.

A common approach of determining one-sided heat transfer coefficients from their overall counterparts that aligns well with the experimental procedure described previously, was published by Wilson [14].

Based on this, many authors proposed modifications to the original approach and extended its application to further heat transfer devices, underlining its meaning as a basic tool for researchers in this field. One of the first and regularly referred modifications was made by Briggs and Young, increasing the number of parameters to be calculated [15], which slightly reduces the need for assumptions. The

Fundamentals general concept bases on the fact, that by changing the fluid flow on one side, only the heat transfer of this side is affected. By keeping the flow of the other side constant, the impact of the modified side can be analyzed. In case of jacketed, stirred tanks, this means modifying the stirring frequency while keeping the jacket flow constant, or vice versa. However, it is prerequisite that there is a functional relation between the characteristic velocity and the heat transfer coefficient of that side to be modified. For variation of the impeller speed, the following well known (simplified) definition can be used:

α ∙ d_v

λ = Nu = C Re ^2⁄3 Pr ^{1⁄3 2-41}

with

Re =ρ Nd ² η

2-42

In the classical approach not only the functional relation must be known, but also its exponents. The mathematical structure then allows setting up a linear equation:

1

U = d_v

C ∙ λ ∙ 1 Re^2⁄3Pr ^1⁄3

+ (s_w λ_w+ 1

α) 2-43

The parameters of the linear regression allow calculation of both, the jacket side heat transfer coefficient, αj, from the intercept and the process side geometrical factor, C, from the slope, respectively (figure 2-5). Further, the intercept represents the extrapolated point, where the process side thermal resistance approximates zero, representing a consistent graphical interpretation of the plot. However, the chart also shows that equidistant spacing of stirring frequencies during experimental determination might lead to unfavorable mathematical weighing of settings that show lower measurement accuracy. For example, in a tubular heat exchanger, higher fluid flow velocities often go along with lower measurable temperature differences and thus, are more impacted by noise.

Fundamentals

Figure 2-5. Example of a Wilson plot applied to a jacketed stirred tank reactor with the aim to determine the jacket side heat transfer coefficient by varying the stirring frequency.

The Wilson plot technique enables a deeper investigation of overall heat transfer coefficients, with very limited additional effort, if certain prerequisites are fulfilled. For stirred tank bioreactors, the required exponential relation between characteristic velocities and their corresponding heat transfer coefficients are available and thus, accessible for this approach. Strictly, the Wilson plot method only is valid for fully established turbulent flow. Further, for heat transfer with phase change, the thermal resistances of both sides cannot be considered independent and thus, further mathematical modifications are needed.

For further reading, the review of Fernández-Seara et al. is recommended [16].

U

[K m

W

]

(Re

Pr

)

[-]

intercept: s_w/l_w + 1/a_j slope

: d^v/(Cl)^r

Fundamentals

2.4 Biothermodynamics on single-cell level

The fundamental feature of thermodynamic considerations is the investigation of a given system within defined boundaries, i.e., the thermodynamic system. A simple system can be characterized by its state variables:

• Pressure (p)

• Specific volume³ (v)

• Absolute temperature (T)

• Specific inner energy (u)

To describe the behavior of the system, the state variables are brought into relation:

Thermal state function

p = p(T v) ^2-44

Energetic state function

u = u(T v) ^2-45

The above equation system says, that if both, temperature and specific volume are given, it must be possible to calculate pressure and inner energy, accordingly.

Furthermore, there are other important state variables and functions that can be used to characterize a system for given applications:

Enthalpy

H = U′ + p ∙ V ^2-46

Differential of the entropy dS =1

T∙ (dU′ + p ∙ dV) ^2-47

Free enthalpy

F′ = U′ − TS ^2-48

Gibbs free enthalpy

G = H − TS ^2-49

From the thermodynamic perspective, there is no difference between living and, to stay in this terms, dead matter. Hence, biological systems can be object to such considerations and the above equations

Fundamentals There are a lot of interesting chemical substances that are difficult to synthesize, because of overall complexity of the molecule structure, chirality or other properties. For some of these products, biotechnology might offer a suitable alternative. However, the overproduction of certain substances usually goes at the expense of other internal pathways. One example is the production of large quantities of lipids, a potential biofuel, in algae [17]. The thermodynamic assessment of metabolic pathways of producing cells has the potential to estimate the feasibility of such processes upfront, i.e., before extensive experimentation. Ultimately, such assessment would even allow estimation of resulting yield coefficients.

The following section summarizes briefly the comprehensive work of [18], giving a sound mathematical foundation to assess biological processes from a thermodynamic perspective.

Figure 2-6. Entropy balance over a living cell, representing an entropy producing, irreversible process. Modified after [19].

For the understanding of thermodynamic processes in living matter it is a reasonable approach to evaluate the entropy balance over a living cell (figure 2-6). As the fundamental principle of live is reproduction, the main purpose of metabolic activity is the formation of highly organized matter from low-complex substrate molecules. Under most cases this would reduce the systems entropy and, hence, would not be a spontaneous process. Other accompanying processes must compensate this by producing entropy on the other side. Further, this net production of entropy must be released to the environment to avoid thermal cell death. This is done either by releasing heat and/ or a large flux of small catabolic waste products. The mathematical formulation of this process is given by equation 2-50.

dS dt= q

T+ ∑ s̅_n∙ ṅ_n− s̅_x∙ ṅ_x+ Ṡ_{p d}

n

2-50