Derivation of the Log Mean Temperature Difference

 

  

Figure 2.1: A Counterflow Heat Exchanger

Using the energy equation to express the heat transferred over a section of a heat exchanger (Figure 2.1)¹:

dQ = . m   dh + h d . m

(2.1)

If the mass flow rate through the heat exchanger is constant, this reduces to:

dQ = . m   dh

(2.2)

For fluids that have constant specific heats and that do not undergo a phase change, a property relation is:

dh = c_p dT

and, therefore:

dQ = . m   cp dT

(2.4)

Using this equation to examine the hot and cold streams separately yields:

 

dQ = - Ch dTh ; Ch = ( . m   cp)h

(2.5)

 

dQ = ±Cc dTc ; Cc = ( . m   cp)c

(2.6)

  

Figure 2.2: Parallel-flow Temperature Variation Over Area

  

Figure 2.3: Counterflow Temperature Variation Over Area

where ± or $\mp$ appears, the top sign designates parallel-flow (Figure 2.2), and the bottom sign designates counterflow (Figure 2.3).

A heat exchanger relationship for expressing the heat transfer between fluids over a differential area is:

(2.7)

Th - Tc = dQ U dA

(2.8)

Rearranging equations 2.5 and 2.6 yields:

dT_h = - dQ C_h
and dT_c = ± dQ C_c
; so

$$\mp$$ dTh - dTc = d(Th-Tc) = dQ (- 1 Ch 1 Cc )

(2.9)

Dividing the above equation by equation 2.8 results in:  

$$\mp$$ d(Th-Tc) Th - Tc = U (- 1 Ch 1 Cc ) dA

(2.10)

The conductive and convective coefficients are assumed to vary only slightly over the heat exchanger area, and the temperature-enthalpy relationship is assumed to be linear, so the overall heat transfer coefficient (U) and the specific heats (C_c and C_h) will be considered to be independent of the area. Equation 2.10 can then be integrated over the heat exchanger:

$$\mp$$ ó
õ side 2

side 1 
1 T_h - T_c
d(T_h-T_c) = U (- 1 C_h
1 C_c
) ó
õ s2

s1 
dA

Parallel-flow:

ln [ T_h2-T_c2 T_h1-T_c1
] = U A (- 1 C_h
- 1 C_c
)

Counterflow:  

ln [ Th2-Tc1 Th1-Tc2 ] = U A (- 1 Ch + 1 Cc )

(2.11)

Returning to equations 2.5 and 2.6, expressions incorporating C_c and C_h can be found:

ó
õ side 2

side 1 
dQ = - ó
õ s2

s1 
C_h dT_h

ó
õ s2

s1 
dQ = ± ó
õ s2

s1 
C_c dT_c

Assuming that the specific heat is also not a function of temperature, this results in:

(2.12)

(2.13)

Replacing the specific heats of equation 2.11 with those found in 2.12 and 2.13 yields

Parallel-flow:

ln [ T_h2-T_c2 T_h1-T_c1
] = U A Q
[ (T_h2 - T_h1) + (T_c1 - T_c2) ]

Counterflow:

ln [ T_h2-T_c1 T_h1-T_c2
] = U A Q
[ (T_h2 - T_h1) + (T_c2 - T_c1) ]

Rearranging to separate Q and UA results in the familiar log mean temperature difference:

Q = U A DT₁ - DT₂

DTLog Mean = DT1 - DT2

(2.14)

It must be remembered that this equation is only applicable under a number of limiting assumptions. The mass flow rate must be constant throughout the heat exchanger. The conductive and the convective coefficients may vary over the heat exchanger area, but only slightly. The enthalpy-temperature relationship must be linear, and the specific heats are considered to be independent of both temperature and area.

From equation 2.14 it is evident that:

for the same inlet and outlet temperatures. In order to better magnify the error resulting from the LMTD, only counterflow heat exchanger configurations will be studied, since that LMTD represents the maximum temperature potential for heat transfer. For counterflow, equation 2.14 is

Q = U A (T_h1 - T_c2) - (T_h2 - T_c1)

This equation is valid only if C_c ¹ C_h; otherwise, T_h1-T_h2 = T_c2 - T_c1 and a logarithm of zero results.

Finally, since the objective of this study is to correctly size heat exchangers, equation 2.14 is rearranged to find the overall conductance multiplied by the area (UA)

 

U A = Q DTLog Mean

(2.15)