Mean temperature difference

It is necessary to find the average value of the temperature difference 9m to be used in the general equation  

The outside stream specific heat Cp and mass flow rate Gt falls in temperature from Tn It) Ti2- 

The inside stream specific heat Cp2 and mass flow rate G2 rises in temperature from T21 to T22. 

Over a small element of area cL4 where the temperatures of the streams are T, and 75, The temperature difference  

Underwood(1) proposed the following approximation for the logarithmic mean temperature difference  

It can be shown that the logarithmic mean temperature difference may also be used for the batchwise heating or cooling of fluids. In such cases, the logarithmic mean of the temperature differences at the beginning and end of the operation should be used as the mean temperature difference. 

If variations of U, and the specific heat(s) and flow rate(s) of the fluid(s) flowing without phase change are negligible, then the relationship between ) and At should be linear. Thus, 

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  

When the condensation process is not exactly isothermal but the temperature change is small such as where there is a significant change in pressure, or where a narrow boiling range multicomponent mixture is being condensed the logarithmic temperature difference can still be used but the temperature correction factor will be needed for multipass condensers. The appropriate terminal temperatures should be used in the calculation. 

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It is thus always possible to achieve the interval design with (S -1) matches and each match operating with the log mean temperature difference of the interval. ... 

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. 

ARm log—mean temperature difference for a given heat exchange... 

The rate of heat-transfer q through the jacket or cod heat-transfer areaM is estimated from log mean temperature difference AT by = UAAT The overall heat-transfer coefficient U depends on thermal conductivity of metal, fouling factors, and heat-transfer coefficients on service and process sides. The process side heat-transfer coefficient depends on the mixing system design (17) and can be calculated from the correlations for turbines in Figure 35a. 

F, Factor, ratio of temperature difference across tube-side film to overall mean temperature difference Dimensionless Dimensionless... 

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

Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Tmn.s. Am. Soc. Mech. Eng., 67,621 (1945)]. 

Care must be taken that does not vaiy too strongly, that the proper equations and conditions are chosen for calculating the individual coefficients, and that the mean temperature difference is the correct one for the specified exchanger configuration. 

Mean Temperature Differenee The temperature difference between the two fluids in the heat exchanger vm, in general, vaiy from point to point. The mean temperature difference (AT, or MTD) can be calculated from the terminal temperatures of the two streams if the following assumptions are valid ... 

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... 

There are two basic approaches to heat-exchanger design for low temperatures (1) the effec tiveness-NTU approach and (2) the log-mean-temperature-difference (LMTD) approach. The LMTD approach is used most frequently when all the required mass flows are known and the area of the exchanger is to be determined. The effec-... 

AT ,[ = Mean temperature difference in °F. For our case it is the log mean temperature differenee. 

AP,rav = Tray pressure drop, inches of liquid ATi = Condensing side temperature difference, °F ATn, = Log mean temperature difference, °F Pf = Foam density, Ibs/ft ... 

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 ... 

Finally, there is an interesting article" that shows how to use Taylor s series to generate shortcut methods from established theory. Examples are given for developing a criterion for replacing log mean temperature differences with average differences and for estimating the effect of temperature on reaction rate. 

Parameters Cj and Cj are specific heats of the cold and hot fluids in Btu respectively, and the log-mean temperature difference is defined as follows ... 

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