= the heat transfer surface available, m2, = the heat transferred, W, = the overall heat transfer coefficient, W/m2K,

A 7 = the temperature difference between the heating or cooling agent and the mass itself, K.

The coefficient U represents the conductivity of the system and it depends on the vessel geometry, fluid properties, flow velocity, wall material and thickness (Scragg, 1991). 1/U is the overall resistance to heat transfer (analogous to 1/K for gas-liquid transfer; Chapter 9). It is the reciprocal of the overall heat-transfer coefficient. It is defined as the sum of the individual resistances to heat transfer as heat passes from one fluid to another and can be expressed as: 111111

= outside film coefficient, W/m2K, = inside film coefficient, W/m2K, = outside fouling film coefficient, W/m2K,

= inside fouling film coefficient, W/m2K, = wall heat transfer coefficient = k/x, W/m2K, k = thermal conductivity of the wall,

There is more detailed discussion of U in Bailey and Ollis (1986), Atkinson and Mavituna (1991b) and Scragg (1991).

where h h h

Atkinson and Mavituna (1991b) have given three methods to determine AT (the temperature driving force) depending on the operating circumstances. If one side of the wall is at a constant temperature, as is often the case in a stirred fermenter, and the coolant temperature rises in the direction of the coolant flow along a cooling coil, an arithmetic mean is appropriate:

Aror

where Tf = the bulk liquid temperature in the vessel,

Te = the temperature of the coolant entering the system,

T) = the temperature of the coolant leaving the system.

If the fluids are in counter- or co-current flow and the temperature varies in both fluids then a log mean temperature difference is appropriate:

where ATC = the temperature of the coolant entering,

A 7] = the temperature of the coolant leaving. If the flow pattern is more complex than either of the two previous situations then the log mean temperature difference defined in equation (7.7) is multiplied by an appropriate dimensionless factor which has been evaluated for a number of heat-exchanger systems by McAdams (1954).

Appropriate techniques have just been discussed to obtain values for <2exch, U and AT (or ATam or ATm). If equation (7.3) is now rearranged:

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