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Additional Curves for Calculating Countercurrent Heat Exchange
Appendix II
Additional Curves for Calculating Countercurrent Heat Exchange I heat transfer equations were given for a sphere uniformly heated or cooled in counterflow. The infinite series involved converge rapidly, as may be seen below. The rapidity of convergence increases with an increase in the Fourier number, and with Fo ^ 0-2 the second and subsequent terms in the series are insignificant by comparison with the first term. Because of this it is possible to write equations of the type IN APPENDIX
0
= 1 - ^ + E Q - S U ^ e x p ( - i f Fo)
(II.l)
in a form suitable for representation in a semi-logarithmic form. This is possible both for m> 1-0 and < 1-0. The equation of the type (II.l) was transposed into the form 00
[l_(l_ w ) 0 ]
=
(1_m) £
i=l
sin v n c
-£exp(-)?Fo).
(II.2)
"
As a result of rapid convergence when Fo > 0-20, and for m > l - 0 and ra< 1-0, the curves corresponding to numerical values of the Biot criterion may be drawn as straight lines. With Fo < 0*20 the lines become curves, all converging at the point [1 — (1 —ra)0]= 1. The plotting of these curves has involved cal culating not merely the first terms of the power series, but also the subsequent ones. For the case when m = 1-0 the basic equations are not trans posed. It was found, by calculation, that when Fo > 0-1 terms other than the first may be disregarded. 254
Calculating Countercurrent Heat Exchange In all, twenty-seven graphs were plotted, and are shown in Figs. II.1-II.9, which cover the majority of cases when the ratio of thermal capacities is 0-1, 0-4, 0-6, 0-8, 1-0, 2-0, 4-0, 6-0 and 8-0. Three graphs were plotted for each value of m: 1. for the heating or cooling of the surface of the sphere; 2. for the heating or cooling of the sphere as a whole; 3. for the heating or cooling of the centre of the sphere; and the value of the expression [1 — (1 —m)0] was determined for each m. The calculation of the bed height at which heat exchange will be accomplished with a particular degree of completeness must be carried out in the sequence given in Appendix I. The values of the expression [1 — (1 — m) 0] for varying degrees of heat exchange will be: For m < 1-0, [l-(l-i»)0'£|=l-/Wo. Form ^ 1-0, m ri n ™\/zi"i ~Pm>i-o [l-(l-m)0M] = m_(1_^>io). The sequence of calculation by these graphs is as in the following example. EXAMPLE
Let us find the coke bed depth for the example of Appendix I if fl^i-o = 0-80 The value of 0 M GM.
F0==oo
FO=~
and
^ < 1 . 0 = 0-90.
is the same in both cases:
= jzr^
= y z ^ g - = 5-0;
but the value [ l - ( l - / w ) 0 M f l ] = l - 0 m < r o = 1-0-80 = 0-20 and [ l - ( l - m ) < 9 ' ^ 2 ] = 1-0-90 = 040. The size of the burden, its properties and the conditions of heat exchange remained the same, and so the values of Bi will also be 255
Appendix II
01 FIG.
0
05
10
1-5
.20 Fo
II 1. Additional graphs for the calculation of heating or cooling in counterflow (m = 0*1). a — surface; b — mean bulk; c — centre of sphere.
256
Calculating Countercurrent Heat Exchange
FIG. II.2. Heating or cooling in counterflow (m = 0-4).
257
Appendix II
FIG. II.3. Heating or cooling in counterflow (jn = 0-6).
258
Calculating Countercurrent Heat Exchange
0-5
■«y^^c 10
15
20 Fo
FIG. II.4. Heating or cooling in counterflow (m = 0*8).
259
Appendix II
260
Calculating Countercurrent Heat Exchange
FIG. II.6. Heating or cooling in counterflow (m = 2-0).
261
Appendix II
OAFo
FIG. II.7. Heating or cooling in counterflow (m = 4-0).
262
Calculating Countercurrent Heat Exchange
(a>
(b)
(c>
i&ismmft. 01
0-2
°*%^ 03
0 4 Fa
FIG. II.8. Heating or cooling in counterflow (m = 60).
263
Appendix II
0
0-1
0-3
0-4
0 5Fo
FIG. II.9. Heating or cooling in counterflow (m = 8-0).
264
Calculating Countercurrent Heat Exchange the same: Bi
M , A
32X0-075 0-4
=6
.Q
For the given values of Bi, [1 — (1 — rri)0'^\ and m = 0-8 we find Fo from the graphs (Fig. II.4): Fo x = KHJ&p = 1-00, Fo 2 = xH2/R2p = 1-43. Hence, the heights H can be determined: R*p F 0 l 0-075»XlOXl-0 i = - ^ — = 0^01675 0-075X1-OX 143 H2 = /L+/~l = 4-80 m. 0-001675
H
, ,. =3'36m'
What, now, will be the temperature of the gases discharged from the layer at these heights, if the heat losses are the same in both cases ? The value [1 — (1 — m) 0'^] corresponds to a degree of complete ness of heat exchange P'm<1.0 = 0-80. This enables us to find the temperature ratio for the lump as a whole on discharge from the top heat transfer stage: _ 1-0-2 Further, by using eqn. (1.17) we can find Q'g and hence the gas discharge temperature t'g', as the initial gas temperature is known (t'g = 160 °C). ®o = 5 r ' j f
= l+m&'^
= 1+(0-8x4-0) = 4-2.
Hence, for
t'M = 1000 °C
we find that
t'g9 = 800 °C.
For a degree of completeness of heat exchange /^
Additional Curves for Calculating Countercurrent Heat Exchange I heat transfer equations were given for a sphere uniformly heated or cooled in counterflow. The infinite series involved converge rapidly, as may be seen below. The rapidity of convergence increases with an increase in the Fourier number, and with Fo ^ 0-2 the second and subsequent terms in the series are insignificant by comparison with the first term. Because of this it is possible to write equations of the type IN APPENDIX
0
= 1 - ^ + E Q - S U ^ e x p ( - i f Fo)
(II.l)
in a form suitable for representation in a semi-logarithmic form. This is possible both for m> 1-0 and < 1-0. The equation of the type (II.l) was transposed into the form 00
[l_(l_ w ) 0 ]
=
(1_m) £
i=l
sin v n c
-£exp(-)?Fo).
(II.2)
"
As a result of rapid convergence when Fo > 0-20, and for m > l - 0 and ra< 1-0, the curves corresponding to numerical values of the Biot criterion may be drawn as straight lines. With Fo < 0*20 the lines become curves, all converging at the point [1 — (1 —ra)0]= 1. The plotting of these curves has involved cal culating not merely the first terms of the power series, but also the subsequent ones. For the case when m = 1-0 the basic equations are not trans posed. It was found, by calculation, that when Fo > 0-1 terms other than the first may be disregarded. 254
Calculating Countercurrent Heat Exchange In all, twenty-seven graphs were plotted, and are shown in Figs. II.1-II.9, which cover the majority of cases when the ratio of thermal capacities is 0-1, 0-4, 0-6, 0-8, 1-0, 2-0, 4-0, 6-0 and 8-0. Three graphs were plotted for each value of m: 1. for the heating or cooling of the surface of the sphere; 2. for the heating or cooling of the sphere as a whole; 3. for the heating or cooling of the centre of the sphere; and the value of the expression [1 — (1 —m)0] was determined for each m. The calculation of the bed height at which heat exchange will be accomplished with a particular degree of completeness must be carried out in the sequence given in Appendix I. The values of the expression [1 — (1 — m) 0] for varying degrees of heat exchange will be: For m < 1-0, [l-(l-i»)0'£|=l-/Wo. Form ^ 1-0, m ri n ™\/zi"i ~Pm>i-o [l-(l-m)0M] = m_(1_^>io). The sequence of calculation by these graphs is as in the following example. EXAMPLE
Let us find the coke bed depth for the example of Appendix I if fl^i-o = 0-80 The value of 0 M GM.
F0==oo
FO=~
and
^ < 1 . 0 = 0-90.
is the same in both cases:
= jzr^
= y z ^ g - = 5-0;
but the value [ l - ( l - / w ) 0 M f l ] = l - 0 m < r o = 1-0-80 = 0-20 and [ l - ( l - m ) < 9 ' ^ 2 ] = 1-0-90 = 040. The size of the burden, its properties and the conditions of heat exchange remained the same, and so the values of Bi will also be 255
Appendix II
01 FIG.
0
05
10
1-5
.20 Fo
II 1. Additional graphs for the calculation of heating or cooling in counterflow (m = 0*1). a — surface; b — mean bulk; c — centre of sphere.
256
Calculating Countercurrent Heat Exchange
FIG. II.2. Heating or cooling in counterflow (m = 0-4).
257
Appendix II
FIG. II.3. Heating or cooling in counterflow (jn = 0-6).
258
Calculating Countercurrent Heat Exchange
0-5
■«y^^c 10
15
20 Fo
FIG. II.4. Heating or cooling in counterflow (m = 0*8).
259
Appendix II
260
Calculating Countercurrent Heat Exchange
FIG. II.6. Heating or cooling in counterflow (m = 2-0).
261
Appendix II
OAFo
FIG. II.7. Heating or cooling in counterflow (m = 4-0).
262
Calculating Countercurrent Heat Exchange
(a>
(b)
(c>
i&ismmft. 01
0-2
°*%^ 03
0 4 Fa
FIG. II.8. Heating or cooling in counterflow (m = 60).
263
Appendix II
0
0-1
0-3
0-4
0 5Fo
FIG. II.9. Heating or cooling in counterflow (m = 8-0).
264
Calculating Countercurrent Heat Exchange the same: Bi
M , A
32X0-075 0-4
=6
.Q
For the given values of Bi, [1 — (1 — rri)0'^\ and m = 0-8 we find Fo from the graphs (Fig. II.4): Fo x = KHJ&p = 1-00, Fo 2 = xH2/R2p = 1-43. Hence, the heights H can be determined: R*p F 0 l 0-075»XlOXl-0 i = - ^ — = 0^01675 0-075X1-OX 143 H2 = /L+/~l = 4-80 m. 0-001675
H
, ,. =3'36m'
What, now, will be the temperature of the gases discharged from the layer at these heights, if the heat losses are the same in both cases ? The value [1 — (1 — m) 0'^] corresponds to a degree of complete ness of heat exchange P'm<1.0 = 0-80. This enables us to find the temperature ratio for the lump as a whole on discharge from the top heat transfer stage: _ 1-0-2 Further, by using eqn. (1.17) we can find Q'g and hence the gas discharge temperature t'g', as the initial gas temperature is known (t'g = 160 °C). ®o = 5 r ' j f
= l+m&'^
= 1+(0-8x4-0) = 4-2.
Hence, for
t'M = 1000 °C
we find that
t'g9 = 800 °C.
For a degree of completeness of heat exchange /^