Log Mean Temperature Difference

Before equation (eq – 1.7) can be used to determine the heat transfer area required for a given duty, an estimate of the mean temperature difference Δtm must be made. This will normally be calculated from the terminal temperature differences: the difference in the fluid temperatures at the inlet and outlet of the exchanger. The well-known “logarithmic mean” temperature difference (LMTD) is only applicable to sensible heat transfer in true co-current or counter current flow. For counter flow the LMTD is given by:

In a heat exchanger the temperatures of the hot and cold fluids keep on changing from point to point along the length of the exchanger. The question therefore arises what value of the temperature difference should be used to compute the rate of heat flow. The need of mean temperature difference which, when multiplied by the overall coefficient of heat transfer and the appropriate area, will give the correct heat flow, originated. An expression for the mean temperature difference

Where ΔT m = log mean temperature difference,
T1 = inlet shell side fluid temperature,
T2 = outlet shell side fluid temperature,
t1 = inlet tube side temperature,
t2 = outlet tube-side temperature,

The equation is the same for co-current flow, but the terminal temperature differences will be (t1 – t1) and (T2 – t2). Strictly, equation (eq – 1.9) will only apply when there is no change in the specific heats, the overall heat-transfer coefficient is constant, and there are no heat losses.

In most shell and tube exchangers the flow will be a mixture of co current, counter current and cross flow. Fig 1.3 show typical temperature profiles for an exchanger with one shell pass and two tube passes (a 1: 2 exchanger).