An actual reactor was not available and tests could not be run to determine the heating and cooling power necessary. Therefore, the heat loss was modeled applying equations found online and in heat transfer textbooks (e.g. [1]). Equations for each surface of the bioreactor were mathematically pieced together to estimate the heat loss for the entire bioreactor. An initial understanding of the bioreactor’s heating requirements were obtained through the procedure described below.

**Hypothetical Bioreactor Operating Conditions**

Apart from knowing the dimensions of the bioreactor’s cylindrically shaped glass walled vessel, parameters useful in approximating the bioreactor’s heating needs, such as mixing rate and fluid level were unknown. Thus, hypothetical operating conditions were set to simulate maximum heat loss from the bioreactor. The maximum heat loss occurs when the bioreactor is full and no components are delivering heat to the system (e.g., growth lights are off); assuming, of course, that the internal temperature is greater than the temperature of the air surrounding the reactor. This was done to estimate the maximum ammount of heat needed to maintain a constant internal temperature inside the reactor. Other assumptions were made to simplify the situation and obtain decent estimate:

- Bioreactor contents are well mixed
- Bioreactor assumed to be in a temperature-controlled environment at 20°C.
- The room/environment around the bioreactor acts as a heat sink at 20°C/68°F
- Natural convection occurs around the bioreactor

The heating power required to maintain different temperatures inside the bioreactor is the not same. Therfore, it was wise to simulate the heat loss at different internal tempeartures. Simulation data provided guidance when selecting compoenents to heat the bioreactor, only components that could produce sufficient heat to overcome the simulated heat loss were considered. Therefore, the heat loss was initially modeled at two hypothetical operating temperatures: (1) a common internal operating temperature (30°C) and (2) an “extreme” operating temperature (45°C).

**Procedure**

To model the total heat loss from the bioreactor, each surface was considered isolated from the rest. In other words, the top, sides, and bottom of the bioreactor were modeled using empirical equations that were derived form experiments on similar geometries: a plate with a hot bottom, a cylinder, and a plate with a hot top, respectively. As mentioned earlier, these equations are found online and in heat transfer textbooks (e.g. [1]). The heat loss from each surface was summed to determine the total heat loss from the bioreactor.

The resistor analogy was used in setting up the calculations for estimating heat loss from the bioreactor: the resistance to heat transfer through multiple materials can be thought of as resistors in series that add up to the total resistance to heat transfer. For example, let’s take a look at finding the resistance to heat transfer through the sides of the reactor. In this case, heat travels from the growth media, through the glass, and into the air. Please follow my hand-written calculations* while reading this procedure. The resistance to heat transfer in the media was considered negligible because the reactor was assumed to be well mixed. Next, the resistance to heat transfer through the bioreactors’ glass wall and into the air, was modeled using equations for conductive heat transfer through a cylinder wall (p. 106 in [1]) and natural convection occurring around the cylindrical surface (p. 326 in [1] or p. 2 of my calculations). Thus, the total resistance to heat transfer is the sum of heat transfer resistances for each material:

R_{side, tot} = Resistance of cylinder glass wall + Resistance of air = ln(r_{o}/r_{i})/2πK_{g}H + (2πr_{o}Hh_{o})^{-1}

Where r_{o} and r_{i} are the outer and inner radi of the cylinder, respectively; H is the height of the reactor; K_{g} is the thermal conductivity of glass; and h_{o} is the convective heat transfer coefficient of air. Variables or material properties such as K_{g} and h_{o} are best found in tables or determined from equations, because lab experiments—outside the scope of most heat transfer projects—must be performed to determine their values.

First, the thermal conductivity of glass, K_{g}, which is 1.4W/mK, was obtained from a popular heat transfer textbook [1]. Then, the resistance to conductive heat transfer of the bioreactor wall was calculated—it is, 0.004 K/W, a value that is negligible in calculations, for an internal bioreactor temp of 30°C and not much greater for 45°C

Second, determining the air’s resistance to heat transfer was more cumbersome. The value for the convective heat transfer coefficient, h_{o}, was obtained using the equation for natural convective flow along the side of a vertical cylinder. This required determining the temperature at the boundary film layer of air, surrounding the cylindrical wall, where convection occurs. The temperature at the boundary film was determined using an iterative process (p. 5 of handwritten steps and p. 553 in [1]). The properties of air, at the boundary-layer temperature, needed to determine h_{o} were interpolated from values in a table found online. The result values for h_{o }and, subsequently, the resistance to heat loss through air, at a hypothetical internal temp of 30°C, were 2.48 W/m^{2}K and 0.91 K/W, respectively. Therefore, the total resistance to heat transfer through the cylindrical bioreactor wall and into the air, R_{side, tot},was 0.91 K/W.

Next, the rate of heat transfer, q, was calculated, where q = ΔT/R_{tot}, for the temperature, ΔT, between the internal bioreactor temp. and the external air temp. (e.g., ΔT = 30-20°C = 10°C or 10°K). The rate of heat loss from the side of the reactor was estimated to be 10.9W, when the media in the reactor is at 30°C.

Similarly, the rate of heat transfer loss from the top and bottom of the reactor were determined to be 0.11 and 0.8W, respectively. The equations used to determine the heat loss from the top of the reactor mathematically represent heat transfer from (1) the media and the bottom of the bioreactor lid (p. 562 in [1]), (2) through the bioreactor lid, and (3) from the top of the lid into the air (p. 551). Similarly, equations for the bottom of the bioreactor were for heat loss from a plate with a hot top (p. 551 in [1]). The sum rate of heat transfer from the top, side, and bottom of the reactor was 11.8W, for a hypothetical internal temp of 30°C.

However, empirical equations are not 100% accurate and accounting for all heat losses from the reactor is impractical. The estimate likely has a 20-40% error or more. Therefore, considering a 30% error, the heat loss from the reactor is predicted to be 15W when the internal temp is 30°C.

Similarly, heat loss form the reactor at an internal temperature of 45°C was estimated to be approximately 61W. Therefore, a heater capable of delivering up to 60W directly to the media should keep the media at a constant temperature for temperatures below 45°C. However, the heating system should also heat the reactor contents to a desired temperature in a reasonable period; the procedure for determining the total heating power for the bioreactor is described, in detail, elsewhere.

**Creating and Excel Model**

The above equations and procedures were used to create an adjustable model¨, in Excel, toquickly obtain estimates of the heat loss from the bioreactor at different operating temperatures. Once parts for the bioreactor were purchased or created, heat loss experiments, described here, were compared to the model. The predicted values from the model were close to the values obtained from experiments, 3.8 vs. 4W, respectively; validating the use of this model during the initial stages of development of the heating system for a simple bioreactor. The excel model, can be used for a range of reactor dimensions, and internal and external temperature, which gives the model applicability when initially developing similar reactors.

**Visualizing Heat Loss Using Energy2D**

Figure 1 – Heat exchange model created using Energy 2D. This visualization shows how natural convection removes heat from the tank. To model heat loss the temperate inside the bioreactor was set at 35°C, a probable maximum inside the bioreactor based on design considerations. The image presented here are modified screen shots for improved reader comprehension.

A visual simulation of the heat loss from the bioreactor was created using Energy2D, a NSF funded open-source software currently in development. As the name implies, heat exchange is simulated in 2D. A two dimensional cross section of the bioreactor was created, with material properties and initial temperatures set for each component. Although Energy2D crashed when simulating the convective heat transfer, a nice visual representation can be seen in the initial seconds of simulated time before the program crashes (Figure 1). Unfortunately, a comparison could not be made of the heat loss through each component in Energy2D vs. that of the model in excel, because Energy2D would crash while running the simulation.

**Conclusion**

The above procedure describes a method to approximate the heating needed to maintain a constant temperature within in a bioreactor, without the ability to perform heat loss tests. This methodology, with a few modifications, can be used to determine the cooling power to maintain the reactor contents at a temperature below the external air temp. Part of the motivation for writing up this work was to make it available for others to follow, improve, or use as a template in the development of a heating/cooling system for their own applications. The procedure has been described for a bioreactor; however, it is applicable for understanding the energy requirements of tanks used for a variety of purposes. Thanks for reading.

**References**

Incropera, Frank P. Dewitt, David B. , ed. *Fundamentals of Heat and Mass Transfer*. 5th ed. New York: Publisher: J. Wiley, 2002.

**Footnotes**

*Note that in my hand-written calculations I go through the procedure with an internal temperature of 60°C, but at the end of the procedure, I actually calculate the value for 30°C and 45°C.

¨If comparing the excel model to my handwritten procedure you will notice that the values are not the same. This is because the height of the bioreactor was incorrectly provided when I first made the model. I corrected the height after I could access and measure the bioreactor height.