Theory:
Constant stirred tank reactors (CSTR) are widely used reactors in the industry. They are used to carry out
reactions that require intense agitation, such as the addition of gaseous reactants in a liquid, a solid
reactant in a liquid, or polymerization reactions. (Rawlings 5) Heat exchange in CSTR reactors is a very
important and well studied division. A highly exothermic reaction or a highly endothermic both require
that heat be taken out or put into the reactor respectively and this is why it is important to study heat
exchange in CSTRs.
Heat is the transfer of energy from one substance to another. There are three types of heat transfers,
conduction, convection, and radiation. Heat conduction is the energy transfer at the molecular level. As
molecules collide and bounce off of each other they exchange energy, the high energy particles loose
energy to the low energy ones. Heat convection is the energy transfer as the bulk fluid moves and
radiation is the transfer of energy without a medium, it does not required molecules or a bulk fluid to be
transferred. In this lab we will be mainly studying conduction and convection. (Bird 266)
The rate of heat conducted depends on the thermal conductivity (k) of a substance, the higher the k
value the easier it is to transfer heat through this substance. Temperature is the measure of energy a
substance holds and heat always transfers from a high temperature region to a low temperature region.
Heat is transferred according to the following law, famously known as Fouriers law. (Bird 266-332)
𝑄/𝐴 = 𝑘
𝑑𝑇
𝑑𝑥
(1)
Q is the Heat transferred
A is the area this heat transfers through
K is the thermal conductivity
T is temperature
x is the distance this heat is transferred through
This law states that the heat flow per unit area is proportional to the temperature decrease dT
over a distance dx. The heat transfer, at a boundary, that takes place between a fluid and a
solid goes through a thin film. This heat transfer is not defined directly by the Fouriers law but is
defined by the Newton’s Law of cooling which is defined as follows: (Bird 266-332)
𝑄 = ℎ𝐴(𝑇0 − 𝑇𝑏 )𝐴
(1a)
Q13 is the heat transferred
h is the heat transfer coefficient
T0 is the temperature of the surface
Tb is the temperature of the bulk fluid
A is the area of heat transfer
In this lab we will be studying the heat transfer through three different regions.
1) Heat transfer across the internal fluid to the wall of the stirred tank
2) Heat transfer across the tank wall
3) Heat transfer from the condensing steam to the tank wall
Since these three regions include the heat transfer through several different mediums a
collective heat transfer coefficients (U) can be derived. U is defined as following for this lab:
𝑈=
1
1
1
+
ℎ𝑖 ℎ𝑜
(2)
ℎ𝑖 is the heat transfers coefficient from the fluid in the CSTR to the tank wall. (W/𝑚2 − 𝐾)
ℎ𝑜 is the heat transfer coefficient from the tank wall to the surrounding steam. (W/𝑚2 − 𝐾)
With the overall heat transfer coefficient, Newton’s law of cooling becomes:
𝑄13 = 𝑈(𝑇1 − 𝑇3 )𝐴
(1b)
U is the overall heat transfer coefficient (accounting for heat resistance of all three boundaries
listed above and described by equation 2) (W/𝑚2 − 𝐾)
𝑇1 is the temperature of the steam surrounding the CSTR (K)
𝑇3 is the temperature of the fluid in the CSTR (K)
A is the heat transfer area (𝑚2 )
Some assumptions that are needed to simplify the heat transfer are that the wall thickness is
thin compared to the tank so area is same for both values and that the tank wall has a very high
k value so it has no resistance to heat transfer.
The following picture can help to understand the derivation of the overall heat transfer
coefficient:
Q
Figure 1: Temperature profile
As the heat transfer from the right to the left it first goes through a thin film with heat transfer
coefficient h0 then it flows through the solid with heat transfer coefficients kscale and kwall finally
comes out on the left side where the heat transfer coefficient is h i. The heat transfer through in
the solid-fluid interface is described the equations:
Q = hiAi(t4 – t5) = h0A0(t1 – t2)
(3)
h0 is the heat transfer coefficient across the solid-fluid boundary where temperature difference
is t1 – t2
hi is the heat transfer coefficient across the solid-fluid boundary where temperature difference
is t4 – t5
The heat transfer across the solids follows the Fourier law and is defined as follows:
Q = Ascalekscale(t2-t3)/xscale = Ascalekscale(t3-t4)/xwall
(4)
Since the heat transfer is same across all the walls at steady state, the x, h, and A values can be
combined to give an overall heat transfer coefficient:
1
𝑈𝑖
1
𝑥𝑠𝑐𝑎𝑙𝑒
𝑖
𝑘𝑠𝑐𝑎𝑙𝑒
=ℎ +
∙𝐴
𝐴𝑖
𝑠𝑐𝑎𝑙𝑒
𝑥
+ 𝑘𝑤𝑎𝑙𝑙 ∙ 𝐴
𝑤𝑎𝑙𝑙
𝐴𝑖
𝑤𝑎𝑙𝑙
+
1
𝐴𝑖
ℎ𝑜 𝐴0
(5)
The amount of heat lost or gained by a substance depends on its heat capacity C.
Q=mC(T-T0)
(6)
This equation determines how much heat is gained or lost by a substance as the temperature
drops or is raised. (Packet)
Heat transfer coefficients can be experimentally determined but there are several correlations
that can estimate these. Some examples are:
Outer Wall Heat Transfer Coefficient Estimation
This estimates the h value on the outside of the vessel wall where steam condensation takes
place.
ℎ𝑜 𝐿𝑇
𝑘
𝐿3 𝜌2 𝑔
= 0.925 (
𝜇𝑀
1/3
)
(6)
Where:
ℎ𝑜 is the heat transfer coefficient through film of condensing steam. (W/𝑚2 − 𝐾
𝐿𝑇 is the vertical length of the tank. (m)
k is the thermal conductivity of the fluid. (W/K-m)
ρ is the density of the fluid. (kg/𝑚3 )
g is the gravitational constant. (m/𝑠 2 )
μ is the viscosity of the fluid. (kg/s-m)
M is the mass rate of steam condensed per wetted perimeter described by:
𝑚
𝑀 = 𝜋𝐷𝑠𝑡
𝑇𝑜
(6a)
𝑚𝑠𝑡 is the mass rate of steam condensation. (kg/s)
Saturated Steam Heat Transfer Coefficient Estimation
ℎ𝑜 = 2960(𝐷𝑇𝑜 /𝑚𝑠𝑡 )1/3
(7)
1. Bird, R. B., Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena. 2nd ed. New
York, NY: Jonh Wiley & Sons, Inc., 2002
Packet, Heat transfer to a Fluid in a Stirred Tank