Logarithmic mean

Log arithmic-Mean Driving Force. As noted eadier, linear operating lines occur if all concentrations involved stay low. Where it is possible to assume that the equiUbrium line is linear, it can be shown that use of the logarithmic mean of the terminal driving forces is theoretically correct. When the overall gas-film coefficient is used to express the rate of absorption, the calculation reduces to solution of the equation... 

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... 

Logarithmic-mean solvent concentration between bulk (kmol solvent)/(kmol liquid) (Ibmol solvent)/(lbmol hquid)... 

Vbm Logarithmic-mean inert-gas concentration (5-262) (kmol inert gas)/(kmol gas) (Ibmol inert gas)/(lbmol gas)... 

In dilute systems the logarithmic-mean insoluble-gas and nonvolatile-hquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). 

D. Rectification in vertical wetted wall column with turbulent vapor flow, Johnstone and Pigford correlation =0.0.328(Wi) Wi P>vP 3000 < NL < 40,000, 0.5 < Ns. < 3 N=, v,.gi = gas velocity relative to R. liquid film = — in film -1 2 " [E] Use logarithmic mean driving force at two ends of column. Based on four systems with gas-side resistance only, = logarithmic mean partial pressure of nondiffusing species B in binary mixture. p = total pressure Modified form is used for structured packings (See Table 5-28-H). 

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... 

One such simphfication was suggested by Wiegand [Trans. Am. Jn.st. Chem. Eng., 36, 679 (1940)], who pointed out that the logarithmic-mean mole fraction of inert gas ysM (oi ysM) is often very nearly equal to the arithmetic mean. Thus, substituflon of the relation... 

Figure (14-5) is a plot of Eq. (14-23) from which the value of Nog can be read directly as a function of mGM/LM and the ratio of concentrations. This plot and Eq. (14-23) are equivalent to the use of a logarithmic mean of terminal driving forces, but they are more convenient because one does not need to compute the exit-liquor concentration X. ... 

The Wiegand approximations of the above integrals in which arithmetic means are substituted for the logarithmic means andxg. are... 

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... 

The temperature difference, At, °E, required to satisfy the basic heat transfer relation Q = UA At is the logarithmic mean to the differences in temperatures at the opposite ends of the paths of flow of the two fluids. The temperature flow paths can be represented as shown in Figures 10-30 and 10-31. 

In most multipass exchangers, a combination of counter-current and co-current flow exists as the fluid flows through alternate passes (see Figure 10-29). The mean temperature is less than the logarithmic mean calculated for counter-cur-rent flow and greater than that based on co-current flow. 

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) ... 

It may be noted that using Underwood s approximation (equation 9.10), the calculated values for the mean temperature driving forces are 41.9 K and 39.3 K for counter- and co-current flow respectively, which agree exactly with the logarithmic mean values. 

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