True Mean Temperature Difference

For multipass construction, the flow is neither counter-current nor co-current. As a consequence, the average driving force is no longer the LMTD. As it turns out, what is called the true mean temperature difference (MTD) is somewhat less than the log-mean  

This is the AT, which when multiplied by UA gives the total rate of heat transfer for a shell-and-tube exchanger  

Fpis a correction factor which is unity for pure co-current or pure counter-current flow  

For pure crossflow to a bank of tubes, or for a mixture of crossflow and counter-current flow, the factor is less than unity  

Note that as Z— 0, F f 1 over the entire range of y l. This special case is applicable for condensing vapors. As long as the vapor is condensing, its temperature stays at the boiling point  

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Though these conditions will not be strictly satisfied in practical heat exchangers, the Ft values obtained from the curves will give an estimate of the true mean temperature difference that is sufficiently accurate for most designs. Mueller (1973) discusses these... 

THE IDEAL AND TRUE MEAN TEMPERATURE DIFFERENCE FOR ONE OR MORE EXCHANGER IN SERIES. 

Here AT , is the true mean temperature difference dependent on the exchanger flow arrangement and degree of fluid mixing within each fluid stream. The inverse of the overall thermal conductance UA is referred to as the overall thermal resistance R , which consists of component resistances in series as shown in Fig. 17.22 as follows. 

The LMTD represents a true mean temperature difference for counterflow and parallelflow arrangements under the idealizations listed below. Thus the LMTD correction factor F rep-... 

In the field of heat transfer, a good example of this category of shortcut design method is the famous F correction factor to correct the log mean temperature difference of shell and tube heat exchangers for deviations from true countercurrent flow. For multipass heat exchangers, the assumptions are ... 

In the basic heat transfer equation it is necessary to use the log mean temperature difference. In Equation 2-4 it was assumed that the two fluids are flowing counter-current to each other. Depending upon the configuration of the exchanger, this may not be true. That is, the way in which the fluid flows through the exchanger affects LMTD. The correction factor is a function of the number of tube passes and the number of shell passes. 

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 ... 

For heat exchangers in true counter-current (fluids flowing in opposite directions inside or outside a tube) or true co-current (fluids flowing inside and outside of a tube, parallel to each other in direction), with essentially constant heat capacities of the respective fluids and constant heat transfer coefficients, the log mean temperature difference may be appropriately applied, see Figure 10-33. ... 

F = MTD correction factor, dimensionless, corrects log mean temperature difference for any deviation from true counter-current flow. 

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 ... 

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). 

AT = true temperature difference, the mean temperature difference for use in the design equation 12.1 ... 

In Section 4.5H it was shown that when the hot and cold fluids in a heat exchanger are in true countercurrent flow or in cocurrent (parallel) flow, the log mean temperature difference should be used. 

In the cases where a multiple-pass heat exchanger is involved, it is necessary to obtain a different expression for the mean temperature difference to use, depending on the arrangement of the shell and tube passes. Considering first the one-shell-pass, two-tube-pass exchanger in Fig. 4.9-2b, the cold fluid in the first tube pass is in counterflow with the hot fluid. In the second tube pass the cold fluid is in parallel flow with the hot fluid. Hence, the log mean temperature difference, which applies to either parallel or counterflow but not to a mixture of both types, as in a 1-2 exchanger, cannot be used to calculate the true mean temperature drop without a correction. 

The mathematical analysis is not covered in this introductory text but for both true co-current and counter-current flow the temperature driving force when properly averaged is found to be the logarithmic mean temperature difference between the fluids ... 

Fixed Tubesheet—The advantage of a fixed tubesheet exchanger is that it is a true countercurrent flow exchanger and no reduction of the log mean temperature difference (LMTD) is required for non-true counter-current flow between the shell-side fluid and the tube-side fluid. However, it s impossible to mechanically clean the shell side, as the bundle cannot be extracted from the shell. When such an exchanger fouls on the tube side, it can easily be cleaned. When it fouls on the shell side, you can throw the entire exchanger away. 

The logarithm to the base 10 may be converted to base e by multiplying by 2.3. Experimental tests indicate that the logarithmic mean-temperature difference is not exact for either streamline or turbulent flow of oils, but no means of easily handling the true case are available. 

The usual practice in the design of shell and tube exchangers is to estimate the true temperature difference from the logarithmic mean temperature by applying a correction factor to allow for the departure from true counter-current flow ... 

The discussions in the literature concentrate on the problem of whether Equations 13 and 14 really reflect activation processes (16, 19) or conformational effects (20, 21) or both (8). It is agreed however that the linearity between In (K/kb) and (1/T) as demanded by Equations 13 and 14 will be fulfilled for the normally accessible temperature range. Equations of the type of 14 may thus be used at least to interpolate data. The quantities (ASf — ASf) and (Aff — AHf) have at least some diagnostic value whatever their true meaning is. For convenience, we treat them here as differences of activation entropies and activation enthalpies, respectively. 

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