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14-524: We assume that a generic heat exchanger has two ends (which we call "A" and "B") at which the hot and cold streams enter or exit on either side; then, the LMTD is defined by the logarithmic mean as follows: where Δ T A is the temperature difference between the two streams at end A , and Δ T B is the temperature difference between the two streams at end B . When the two temperature differences are equal, this formula does not directly resolve, so

28-614: A cross-flow, in which one system, usually the heat sink, has the same nominal temperature at all points on the heat transfer surface, a similar relation between exchanged heat and LMTD holds, but with a correction factor. A correction factor is also required for other more complex geometries, such as a shell and tube exchanger with baffles. Assume heat transfer is occurring in a heat exchanger along an axis z , from generic coordinate A to B , between two fluids, identified as 1 and 2 , whose temperatures along z are T 1 ( z ) and T 2 ( z ) . The local exchanged heat flux at z

42-437: Is clearly the pipe length, which is distance along z , and D is the circumference. Multiplying those gives Ar the heat exchanger area of the pipe, and use this fact: In both integrals, make a change of variables from z to Δ T : With the relation for Δ T (equation 1 ), this becomes Integration at this point is trivial, and finally gives: from which the definition of LMTD follows. A related quantity,

56-1083: Is defined as: for the positive numbers x, y . The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean , respectively. The inequalities are strict unless both numbers are equal. 2 1 x + 1 y ≤ x y ≤ x − y ln ⁡ x − ln ⁡ y ≤ x + y 2 ≤ ( x 2 + y 2 2 ) 1 / 2  for all  x > 0  and  y > 0. {\displaystyle {\frac {2}{\displaystyle {\frac {1}{x}}+{\frac {1}{y}}}}\leq {\sqrt {xy}}\leq {\frac {x-y}{\ln x-\ln y}}\leq {\frac {x+y}{2}}\leq \left({\frac {x^{2}+y^{2}}{2}}\right)^{1/2}\qquad {\text{ for all }}x>0{\text{ and }}y>0.} Toyesh Prakash Sharma generalizes

70-455: Is proportional to the temperature difference: The heat that leaves the fluids causes a temperature gradient according to Fourier's law : where k a , k b are the thermal conductivities of the intervening material at points A and B respectively. Summed together, this becomes where K = k a + k b . The total exchanged energy is found by integrating the local heat transfer q from A to B : Notice that B − A

84-702: Is the arithmetic logarithmic geometric mean inequality. Similarly, one can also obtain results by putting different values of n as below For n = 1 : x y ( 1 + ln ⁡ x y ) ≤ x ln ⁡ x − y ln ⁡ y ln ⁡ x − ln ⁡ y ≤ x ( 1 + ln ⁡ x ) + y ( 1 + ln ⁡ y ) 2 {\displaystyle {\sqrt {xy}}\left(1+\ln {\sqrt {xy}}\right)\leq {\frac {x\ln x-y\ln y}{\ln x-\ln y}}\leq {\frac {x(1+\ln x)+y(1+\ln y)}{2}}} for

98-1454: The area under an exponential curve . L ( x , y ) = ∫ 0 1 x 1 − t y t   d t = ∫ 0 1 ( y x ) t x   d t = x ∫ 0 1 ( y x ) t d t = x ln ⁡ y x ( y x ) t | t = 0 1 = x ln ⁡ y x ( y x − 1 ) = y − x ln ⁡ y x = y − x ln ⁡ y − ln ⁡ x {\displaystyle {\begin{aligned}L(x,y)={}&\int _{0}^{1}x^{1-t}y^{t}\ \mathrm {d} t={}\int _{0}^{1}\left({\frac {y}{x}}\right)^{t}x\ \mathrm {d} t={}x\int _{0}^{1}\left({\frac {y}{x}}\right)^{t}\mathrm {d} t\\[3pt]={}&\left.{\frac {x}{\ln {\frac {y}{x}}}}\left({\frac {y}{x}}\right)^{t}\right|_{t=0}^{1}={}{\frac {x}{\ln {\frac {y}{x}}}}\left({\frac {y}{x}}-1\right)={}{\frac {y-x}{\ln {\frac {y}{x}}}}\\[3pt]={}&{\frac {y-x}{\ln y-\ln x}}\end{aligned}}} The area interpretation allows

112-484: The logarithmic mean pressure difference or LMPD , is often used in mass transfer for stagnant solvents with dilute solutes to simplify the bulk flow problem. Logarithmic mean In mathematics , the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient . This calculation is applicable in engineering problems involving heat and mass transfer . The logarithmic mean

126-516: The LMTD is conventionally taken to equal its limit value, which is in this case trivially equal to the two differences. With this definition, the LMTD can be used to find the exchanged heat in a heat exchanger: where (in SI units ): Note that estimating the heat transfer coefficient may be quite complicated. This holds both for cocurrent flow, where the streams enter from the same end, and for countercurrent flow, where they enter from different ends. In

140-2055: The arithmetic logarithmic geometric mean inequality for any n belongs to the whole number as x y   ( ln ⁡ x y ) n − 1 ( n + ln ⁡ x y ) ≤ x ( ln ⁡ x ) n − y ( ln ⁡ y ) n ln ⁡ x − ln ⁡ y ≤ x ( ln ⁡ x ) n − 1 ( n + ln ⁡ x ) + y ( ln ⁡ y ) n − 1 ( n + ln ⁡ y ) 2 {\displaystyle {\sqrt {xy}}\ \left(\ln {\sqrt {xy}}\right)^{n-1}\left(n+\ln {\sqrt {xy}}\right)\leq {\frac {x(\ln x)^{n}-y(\ln y)^{n}}{\ln x-\ln y}}\leq {\frac {x(\ln x)^{n-1}(n+\ln x)+y(\ln y)^{n-1}(n+\ln y)}{2}}} Now, for n = 0 : x y ( ln ⁡ x y ) − 1 ln ⁡ x y ≤ x − y ln ⁡ x − ln ⁡ y ≤ x ( ln ⁡ x ) − 1 ln ⁡ x + y ( ln ⁡ y ) − 1 ln ⁡ y 2 x y ≤ x − y ln ⁡ x − ln ⁡ y ≤ x + y 2 {\displaystyle {\begin{array}{ccccc}{\sqrt {xy}}\left(\ln {\sqrt {xy}}\right)^{-1}\ln {\sqrt {xy}}&\leq &{\dfrac {x-y}{\ln x-\ln y}}&\leq &{\dfrac {x(\ln x)^{-1}\ln x+y(\ln y)^{-1}\ln y}{2}}\\[4pt]{\sqrt {xy}}&\leq &{\dfrac {x-y}{\ln x-\ln y}}&\leq &{\dfrac {x+y}{2}}\end{array}}} This

154-1056: The easy derivation of some basic properties of the logarithmic mean. Since the exponential function is monotonic , the integral over an interval of length 1 is bounded by x and y . The homogeneity of the integral operator is transferred to the mean operator, that is L ( c x , c y ) = c L ( x , y ) {\displaystyle L(cx,cy)=cL(x,y)} . Two other useful integral representations are 1 L ( x , y ) = ∫ 0 1 d t t x + ( 1 − t ) y {\displaystyle {1 \over L(x,y)}=\int _{0}^{1}{\operatorname {d} \!t \over tx+(1-t)y}} and 1 L ( x , y ) = ∫ 0 ∞ d t ( t + x ) ( t + y ) . {\displaystyle {1 \over L(x,y)}=\int _{0}^{\infty }{\operatorname {d} \!t \over (t+x)\,(t+y)}.} One can generalize

SECTION 10

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168-964: The logarithm. For n = 2 this leads to The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex S {\textstyle S} with S = { ( α 0 , … , α n ) : ( α 0 + ⋯ + α n = 1 ) ∧ ( α 0 ≥ 0 ) ∧ ⋯ ∧ ( α n ≥ 0 ) } {\displaystyle S=\{\left(\alpha _{0},\,\dots ,\,\alpha _{n}\right):\left(\alpha _{0}+\dots +\alpha _{n}=1\right)\land \left(\alpha _{0}\geq 0\right)\land \dots \land \left(\alpha _{n}\geq 0\right)\}} and an appropriate measure d α {\textstyle \mathrm {d} \alpha } which assigns

182-401: The mean to n + 1 variables by considering the mean value theorem for divided differences for the n -th derivative of the logarithm. We obtain where ln ⁡ ( [ x 0 , … , x n ] ) {\displaystyle \ln \left(\left[x_{0},\,\dots ,\,x_{n}\right]\right)} denotes a divided difference of

196-436: The proof go through the bibliography. From the mean value theorem , there exists a value ξ in the interval between x and y where the derivative f ′ equals the slope of the secant line : The logarithmic mean is obtained as the value of ξ by substituting ln for f and similarly for its corresponding derivative : and solving for ξ : The logarithmic mean can also be interpreted as

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