The overall heat transfer coefficient for a shell and tube heat exchange for clean surfaces is U_o = 400 W/m²K. The fouling factor after one year of operation is found to be \(\frac{1}{{2000}}\frac{m^2K}{{{}W}}\). The overall heat transfer coefficient at this time is

Explanation:

Overall heat transfer coefficient (OHTC):

• OHTC is defined as the quantity of rate of heat transfer takes place due to any mode per unit area for unit temperature difference.
• It doesn’t have any physical meaning, it’s an experimentally determined coefficient value.
• The concept of OHTC is valid for a steady-state. 
• In the case of a heat exchanger, there is a combined mode of heat transfer or multiple resistances present therefore we use,

​\(Q = \;UA{\bf{\Delta }}T \Rightarrow UA = \frac{1}{{{R_{Total}}}}\), 

where R Total is total resistance and U = OHTC

• Unit of ‘U’ is W/m2K
• U indicates the quantity of rate of heat transfer takes place due to any mode per unit area for the unit temperature difference.

Fouling factor indicates the resistance offered due to chemical deposits and scaling.

So, \(\frac{1}{{{U_{with\;fouling}}}} = \frac{1}{{{U_{without\;fouling}}}} + F\)

Calculation:

Given:

Uo = 400 W/m2K, \(F=\frac{1}{{2000}}\frac{m^2K}{{{}W}}\)

\(\begin{array}{l} \therefore \frac{1}{{{U_{with\;fouling}}}} = \frac{1}{{400}} + \frac{1}{{2000}}\\ \Rightarrow {U_{with\;fouling}} = 333\frac{W}{{{m^2}k}} \end{array}\)