Theory CSTR Heat Transfer 2

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 reaction both require that heat be taken out or put into the reactor respectively.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. This constant is a measure of a substances resistance to heat conduction. 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, known as Fouriers law. (Bird 266-332) (1)Q is the Heat transferredA is the area this heat transfers throughK is the thermal conductivityT is temperaturex is the distance this heat is transferred throughThis 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 Newtons Law of cooling which is defined as follows: (Bird 266-332)(1a)Q13 is the heat transferredh is the heat transfer coefficientT0 is the temperature of the surfaceTb is the temperature of the bulk fluidA is the area of heat transferIn 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 tank2) Heat transfer across the tank wall 3) Heat transfer from the condensing steam to the tank wallSince these three regions include the heat transfer through several different mediums a collective heat transfer coefficients (U) can be derived. U is defined as the following for this lab:(2) is the heat transfers coefficient from the fluid in the CSTR to the tank wall. (W/ is the heat transfer coefficient from the tank wall to the surrounding steam. (W/With the overall heat transfer coefficient, Newtons law of cooling becomes:(1b)U is the overall heat transfer coefficient (accounting for heat resistance of all three boundaries listed above and described by equation 2) (W/ is the temperature of the steam surrounding the CSTR (K) is the temperature of the fluid in the CSTR (K)A is the heat transfer area ()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:

QFigure 1: Temperature profileAs 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 hi. 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 t2hi is the heat transfer coefficient across the solid-fluid boundary where temperature difference is t4 t5The 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 the same across all the walls at steady state, the x, h, and A values can be combined to give an overall heat transfer coefficient: (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 theoretically estimated using the following correlations:Outer Wall Heat Transfer Coefficient EstimationThis estimates the h value on the outside of the vessel wall where steam condensation takes place. (6)Where: is the heat transfer coefficient through film of condensing steam. (W/ is the vertical length of the tank. (m)k is the thermal conductivity of the fluid. (W/K-m) is the density of the fluid. (kg/)g is the gravitational constant. (m/) is the viscosity of the fluid. (kg/s-m)M is the mass rate of steam condensed per wetted perimeter described by:(6a) is the mass rate of steam condensation. (kg/s)

Saturated Steam Heat Transfer Coefficient (7)

Estimation Inner Wall Heat Transfer Coefficient EstimationThis system falls into the category of an unbaffled CSTR with Newtonian fluid. The heat transfer correlation can be estimated using the following correlations.

(3)(9)

Where: is the heat transfer coefficient of the inner CSTR tank wall. (W/ is the diameter of the tank (m) is the viscosity of the fluid. (kg/s-m) is the specific heat of the reactor fluid (J/kg-K)k is the thermal conductivity of the fluid. (W/K-m) is the density of the reactor fluid. (kg/) is the diameter of the impeller. (m)n is the rate of revolution of the impeller. (RMP)

Correlation (3) can also be expressed in terms of dimensionless numbers:(4)Where: is the tank Nusselt number described by:(4a) is the impeller Reynolds number described by:(4b) is the fluid Prandtl number described by:(4c)In this lab the convective heat transfer coefficient at the inner surface of the tank will be experimentally determined. This value will then be compared with theoretically calculated values using the equations listed above.

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