Heat and Mass Transfer 2(2+0)

Lesson-27 Logarithmic mean temperature difference for counter and cross flow heat exchangers, Fouling or scaling of heat exchanger and Numerical problems

θ_i and θ_o represent temperature difference between hot and cold fluids at inlet and outlet of heat exchanger respectively and are expressed as

θ_i = T_hi –T_co and θo = T_ho –T_ci                                                                              (1)

C_ph  and C_pc  are heat capacity rates of hot and cold fluid respectively.

Heat capacity rate of a fluid is defined as amount of heat required to increase temperature of a fluid by 1 ^oC and is expressed as

C_ph = m_h C_{ph, ,} C_pc = m_c C_pc                                                                                    (2)

Consider an element of area dA and thickness dx at a distance ‘x’ from inlet of heat exchanger. Let T_h and T_c are temperatures of hot and cold fluid at inlet of the element and θ represents temperature difference of hot and cold fluid at inlet of the element.

θ = T_h –T_c                                                                                                      (3)

Let dT_h and dT_c represent change in temperature of hot and cold fluids respectively during the flow through the element. Temperature difference between hot and cold fluid at inlet of element is expressed as

dθ = dT_h - dT_c                                                                                                   (4)

During flow through the element, heat transferred is equal to heat lost by hot fluid and heat gained by cold fluid.

If dQ is the amount of heat transferred during flow of fluids through the element then

dQ = UdA θ = -m_c c_pc dT_c =  - m_h c_ph dTh                                                                 (5)

 (-ve sign if temperature of fluid decreases along the length of heat exchanger)

Using equation (2), equation (5) can be written as

dQ = UdA θ = -C_c dT_c =  - C_h dTh                                                                         (6)       

From equation (6), we can write

Substituting the values of dT_c and dT_h  from equation (7) in equation (4), we get

Integrating equation (8) between limits θ_i and θ_o­      

Heat lost by hot fluid during flow through heat exchanger is expressed as

Q =  m_h c_ph (T_hi – T_ho) = C_h (T_hi – T_ho)

Heat gained by hot fluid during flow through heat exchanger is expressed as

Q =  m_c c_pc (T_co – T_ci) = C_c (T_co – T_ci)

Substituting values of C_h and C_c from equations (10) and (11) in equation (9), we get

Using equation (1), equation (12) can be written as

We know that in a heat exchanger, heat transfer between hot and cold fluids can be expressed as

Q = U A (ΔT)_m                                                                                  (14)

Comparing equations (13) and (14), we can write

Equation (15) represents logarithmic mean temperature difference for a counter flow heat exchanger.

C. Cross Flow Heat Exchanger

In order to determine logarithmic mean temperature difference for a counter flow heat exchanger, following relationship is used.

(LMTD)_cross =  F Χ (LMTD)_counter

Where ‘F’ is a correction factor and is a function of geometry of heat exchanger and inlet as well as outlet temperatures of both the fluids. Values of correction factor for different arrangements of cross flow and multipass shell and tube heat exchangers are given in Figure 3.

Fouling Factor