A general expression for U can be easily obtained as follows. Consider a double pipe heat exchanger in which one fluid through the inner pipe and the other fluid through the annular space between space between the two pipes.

Let       L          = length of heat exchanger, m

            r_i          = inside radius of inner pipe, m

            r_o          = outside radius of inner pipe, m

            A_i         = inside surface area of inner pipe (2πr_iL), m².

            A_o        = outside surface area of inner pipe (2πr_oL), m².

            h_i          = film coefficient of heat transfer at inside surface of inner pipe, W/m² C

            h_o         = film coefficient of heat transfer at outside surface of inner pipe, W/m² C

            k_w        = thermal conductivity of inner pipe wall, W/m C.

            t_i          = temperature of fluid flowing through the inner pipe, C

t_o          = temperature of fluid flowing through the annular space between the two  pipes, C

R_i         = thermal resistance of fluid film at the inside surface of inner pipe, m² C/W

R_o        = thermal resistance of fluid film at the outside surface of inner pipe, m² C/W     

R_w        = thermal resistance of inner pipe, m² C/W

(i)         The rate of heat transfer between the two fluids is given by:

                        q          = t_i - t_o                        (eq 1.15)

                                         ΣR

Where              ΣR       = R_i + R_o +R_w               (eq. 1.16)

Since                R_i         =   1/A_ih_i                     (eq. 1.17)

                        R_w        = ln (r_o/r_i)                     (eq. 1.19)

                                        2πLK_w

                        R_o        = 1/A_oh_o                      (eq. 1.20)

Hence

                        q          = .          (t_i - t_o)                           (eq. 1.21)

                                       1/A_ih_i + ln (r_o/r_i) + 1/A_oh_o

                                                     2πLK_w

(ii) If U_i and U_o denote respectively the overall heat transfer coefficient based on unit area of the inside and outside surfaces of the inner pipe, then

                        q          = A_iU_i (t_i - t_o)  = A_oU_o (t_i - t_o)        (eq. 1.22)

from eq. 1.21 and 1.22  U_i       = .                          1                         (eq. 1.23)

                                                1/h_i + A_i ln (r_o/r_i) + [A_i/A_o]. 1/h_o

                                                                2πLK_w

                        U_o        = .                          1                                  (eq. 1.24)

                                       [A_o/A_i].1/h_i + [A_o/2πL].ln (r_o/r_i) + 1/h_o

                                                                                  K_w

(iii) Since A_i = 2πr_iL and A_o = 2πr_oL, eq. c and d can also be written as:

            U_i         = .                          1                             (eq. 1.25)

                           1/h_i + [r_i/k_w] ln (r_o/r_i) + [r_i/r_o]. 1/h_o

Uo       = .                          1                              (eq. 1.26)

                 [r_o/r_i].1/h_i + [r_o/ K_w].ln (r_o/r_i) +  1/h_o

Page: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20

Introduction | Combined heat transfer process | Heat transfer in cooling tower | Variables affecting performance of CT heat transfer | Heat transfer within cooling system (heat exchanger) | Types of heat exchanger | Basic design procedure and theory | Designing a test heat exchanger | Log Mean Temperature difference | L.M.T.D. Correction factors | Overall heat transfer coefficient | Elaborated method for calculating U values | Effect of scale formation | Condensation of steam | Condenser, where the hot fluid temperature varies | Significance of pressure | Significance of flow rate | Methods of checking steam condenser performance | Common conversion factors