Heat exchanger: Logarithmic temperature average
Why log scale?
The most fundamental issue in a shell and tube heat exchanger is that the delta t across cold and hot fluids varies at every point due to changes in fluid thermal properties and metal thermal conductivity. At each point, the fluid and metal properties differ. Temperature increases specific heat and thermal conductivity. Over time, they evolve into non-linear changes. This is the fundamental reason why a log temperature scale is used to calculate the average. The log scale is used by LMTD to linearize the nonlinear thermal properties of fluids and metal.
LMTD
The logarithmic mean temperature difference (LMTD) is used to calculate the temperature driving force for heat transfer in flow systems, most notably in heat exchangers.
LMTD = [delta TA - delta TB]/ln[delta TA/delta TB]
= [delta TA - delta TB]/[ ln delta TA - ln delta TB]
In this case, TA denotes the temperature difference between two streams at end A, and TB denotes the temperature difference between two streams at end B. When the two temperature differences are equal, this formula does not directly resolve, so the LMTD is traditionally taken to equal its limit value, which is trivially equal to the two differences in this case.
The LMTD can be used to calculate the heat exchanged in a heat exchanger using this definition:
Q = U A LMTD
Where Q represents the exchanged heat [in watts], U represents the heat transfer coefficient [in watts per kelvin per square meter], and A represents the exchange area.
This is true for both cocurrent flows, where the streams enter from the same end, and counter-current flow, where the streams enter from opposite ends.
The larger the LMTD for a given heat exchanger with the constant area and heat transfer coefficient, the more heat is transferred. The application of the LMTD stems directly from the analysis of a heat exchanger with a constant flow rate and fluid thermal properties.
Limitations of LMTD
LMTD is valid under the following conditions
- Fluids with constant specific heat, however specific heat changes
- The heat transfer coefficient (U) is constant, and not a function of temperature. This does change in actual life.
- The LMTD is a steady-state concept.
, -No phase change during heat transfer
- Changes in kinetic energy and potential energy are neglected
Why log scale?
The two convective heat transfer coefficients and the conductive thermal resistance of metal vary in the heat exchanger from point to point in a non-linear manner with temperature due to changes in the thermal properties of fluids and metal. The data are linearized using a log scale.
What is nonlinear vs linear change?
A linear function forms a straight line when it is plotted on a graph, and a nonlinear function does not form a straight line (it is curved in some way). The slope of a linear function is constant, whereas the slope of a nonlinear function is continuously changing. Linear changes are predictable while non-linear changes are unpredictable.
The most basic definition of LMTD is a method for converting non-linear regression unpredictive temperature values to linear regression predictive temperature values. Linear regression fits a straight line or surface that minimizes the discrepancies between predicted and actual output values. Nonlinear data are not proportional between two variables. The log scale is a type of exponential scale. Nonlinear thermal properties are linearized using log scale. Logarithmic transformation is a convenient means of transforming a highly skewed variable into a more normalized dataset. When the temperature values are linearized, for a fixed change in the independent variable there is a corresponding fixed change in the dependent variable. This is the fundamental point of what the log scale does
Nonlinear regression to linear regression
When the temperature values are linearized, for a fixed change in the independent variable there is a corresponding fixed change in the dependent variable. This is the fundamental point of what the log scale does