Logarithmic mean-temperature difference

The LMTD, ie, logarithmic mean temperature difference, is an effective overall temperature difference between the two fluids for heat transfer and is a function of the terminal temperature differences at both ends of the heat exchanger. 

At, , A(, Arithmetic- and logarithmic-mean temperature difference respectively K OF... 

Countercurrent or Cocurrent Flow If the flow of the streams is either completely countercurrent or completely cocurrent or if one or both streams are isothermal (condensing or vaporizing a pure component with negligible pressure change), the correct MTD is the logarithmic-mean temperature difference (LMTD), defined as... 

The rate of temperature drop of a fluid as it flows along the length of a heat exchanger is not constant. In order to take account of this nonlinear relationship, the logarithmic mean temperature difference (EMTD) is used. If the inlet and outlet temperatures do not differ widely, an arithmetic mean can be used, because the relationship is considered to be linear. 

The logarithmic mean temperature difference is defined when AT, A7, Consider the case where AT2 = ATj. The logarithmic temperature difference is obtained by applying I lTopital s rule as AT2 —> AT, giving... 

The logarithmic mean temperature difference is the same as the temperature difference at the entrance and exit of the heat exchanger, i.e., AT, = AT, = AT ... 

Note that the logarithmic mean temperature difference should he used when the following conditions generally apply for conditions of true counter-current or co-current flow ... 

GTD = Greater Terminal Temperature Difference, °F LTD = Lesser Terminal Temperature Difference, °F LMTD = Logarithmic Mean Temperature Difference, °F = Tj = Inlet temperature of hot fluid, °F Tj = Outlet temperature of hot fluid, °F tj = Inlet temperature of cold fluid, °F q = Outlet temperature of cold fluid, °F... 

In applying the correlation, use is made of the concept of logarithmic mean temperature difference across the boundary layer. For a boiler section, or pass, this is given by ... 

The value of T is calculated from the logarithmic mean temperature difference multiplied by a correction factor. With single-pass operation, this factor is about 1 except for plate packs of less than 20, when the end effect has a... 

Example 1.11 A fluid evaporates at 3°C and cools water from 11.5°C to 6.4°C. What is the logarithmic mean temperature difference and what is the heat transfer if it has a surface area of 420 m and the thermal transmittance is 110 W/ (m K) ... 

Figure 9.71. Correction for logarithmic mean temperature difference for single shell pass exchanger...

Underwood, A.J.V. Industrial Chemist, 9 (1933) 167. Graphical computation of logarithmic mean temperature difference. 

F Geometric factor for radiation or correction factor for logarithmic mean temperature difference ... 

Before equation 12.1 can be used to determine the heat transfer area required for a given duty, an estimate of the mean temperature difference A Tm 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 (see Volume 1, Chapter 9) is only applicable to sensible heat transfer in true co-current or counter-current flow (linear temperature-enthalpy curves). For counter-current flow, Figure 12.18a, the logarithmic mean temperature is given by ... 

The value of F, will be close to one when the terminal temperature differences are large, but will appreciably reduce the logarithmic mean temperature difference when the temperatures of shell and tube fluids approach each other it will fall drastically when there is a temperature cross. A temperature cross will occur if the outlet temperature of the cold stream is greater than the inlet temperature of the hot stream, Figure 12.18c. 

A pure, saturated, vapour will condense at a fixed temperature, at constant pressure. For an isothermal process such as this, the simple logarithmic mean temperature difference can be used in the equation 12.1 no correction factor for multiple passes is needed. The logarithmic mean temperature difference will be given by ... 

Where integral condensation can be considered to occur, the use of a corrected logarithmic mean temperature difference based on the terminal temperatures will generally give a conservative (safe) estimate of the mean temperature difference, and can be used in preliminary design calculations. 

When the fluid being vaporised is a single component and the heating medium is steam (or another condensing vapour), both shell and tubes side processes will be isothermal and the mean temperature difference will be simply the difference between the saturation temperatures. If one side is not isothermal the logarithmic mean temperature difference should be used. If the temperature varies on both sides, the logarithmic temperature difference must be corrected for departures from true cross- or counter-current flow (see Section 12.6). 

Although the result in Equation 15.45 applies to both countercurrent and cocurrent flow, in practice, cocurrent flow is almost never used as, given fixed fluid inlet and outlet temperatures, the logarithmic mean temperature difference for countercurrent flow is always larger. This in turn leads to smaller surface area requirements. Also, as shown in Figure 15.6a for countercurrent flow, the final temperature of the hot fluid can be lower than the final temperature of the cold fluid (sometimes known as temperature cross), whereas in Figure 15.6b, it is clear that there can never be a temperature cross. 

U = overall heat transfer coefficient Aexisting = existing heat transfer area A A = additional area requirement ATim = logarithmic mean temperature difference Fj = logarithmic mean temperature difference correction factor... 

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