6.1 Overall Heat Transfer Coefficient

Refer to Fig.(15). 


\[Q = \frac{\Delta T}{R} = UA\Delta T = U_1A_1\Delta T = U_2A_2\Delta T\] where \(U\) is the overall heat transfer coefficient, based on the heat transfer area \(A\).

Total resistance (\(R\)) to heat transfer is the sum of resistance offered by inside fluid film, wall resistance, outside fluid film.

From the relation \(Q=U_1A_1\Delta T\), we get \[\frac{Q}{U_1A_1} = Q\left[\frac{1}{h_1A_1} + \frac{x_w}{k_wA_m} + \frac{1}{h_2A_2} \right]\] i.e., \[\frac{1}{U_1} = \frac{1}{h_1} + \frac{x_w}{k_w}\frac{A_1}{A_m} + \frac{1}{h_2}\frac{A_1}{A_2}\] Similarly, we can also write \[\frac{1}{U_2} = \frac{1}{h_2} + \frac{x_w}{k_w}\frac{A_2}{A_m} + \frac{1}{h_1}\frac{A_2}{A_1}\]

For thin-walled tubes, where \(A_1\approx A_2\approx A_m\), we can write, \[\frac{1}{U_1} = \frac{1}{U_2} = \frac{1}{U} = \frac{1}{h_1} + \frac{x_w}{k_w} + \frac{1}{h_2}\] For the case of highly conducting wall, we can write the above as, \[\frac{1}{U} = \frac{1}{h_1} + \frac{1}{h_2}\] The smaller heat transfer coefficient is the controlling film coefficient. Because it offers more resistance. i.e., if \(h_1\ll h_2\), then fluid 1 is called as the controlling film. \[\frac{1}{U} = \frac{1}{h_1}\]

Fouling Factor

Over a time period of heat exchanger operation the surface of the heat exchanger may be coated by the various deposits present in the flow system. These deposits are known as scales. These scales provide another resistance and usually decrease the performance of the heat exchangers. The overall effect is usually represented by dirt factor or fouling factor, or fouling resistance, \(R_f\) which must be included for the calculation of overall heat transfer coefficient. \[R_f = \frac{1}{U_{\text{dirty}}} - \frac{1}{U_{\text{clean}}}\] Thus to determine the \(R_f\), it is very important to know \(U_{\text{clean}}\) for the new heat exchanger. The \(U_{\text{clean}}\) data must be kept securely to obtain the \(R_f\), at any time of the exchanger’s life.

The fluid velocity and the fluid temperature appear to be among the factors that affect the rate of fouling on a given surface. An increase in the velocity decreases both the rate of deposit and the ultimate amount of deposit on the surface. Increasing the fluid bulk temperature increases both the rate of buildup of fouling and its ultimate stable level.

Overall Heat Transfer Coefficient including Fouling Coefficients

\[\frac{1}{U_1A_1} = \frac{1}{h_1A_1} + \frac{1}{h_{1,d}A_1} + \frac{x_w}{k_wA_m} + \frac{1}{h_{2,d}A_2} + \frac{1}{h_2A_2}\] where \(h_{1,d}\) and \(h_{2,d}\) are fouling coefficients.