Pipe and device for thermally cleaving hydrocarbons

Abstract

The invention relates to a pipe for thermal cracking of hydrocarbons in the presence of steam, in which the feed mixture is guided through externally heated pipes, wherein the pipe extends along a longitudinal axis and has a number N.sub.T of grooves that have been introduced into the inner surface of the pipe and extend in a helix around the longitudinal axis along the inner surface, the inner surface into which the grooves have been introduced, in a cross section at right angles to the longitudinal axis, has a diameter Di and a radius r.sub.1=Di/2, the grooves in the cross section at right angles to the longitudinal axis, in their groove base, each have the form of a circular arc and the circular arc has a radius r.sub.2, and
the grooves each have a groove depth TT which, in the cross section at right angles to the longitudinal axis, corresponds in each case to the smallest distance between the circle having the diameter Di on which the inner surface lies and the center of which lies on the longitudinal axis, and the furthest removed point of the groove base of the grooves from the longitudinal axis.

Claims

1. A pipe for thermal cracking of hydrocarbons in the presence of steam, in which a feed mixture is guided through externally heated pipes, wherein the pipe (1) extends along a longitudinal axis (A) and has a number N.sub.T of grooves (2) that have been introduced into the inner surface of the pipe (1) and extend in a helix around the longitudinal axis (A) along the inner surface, the inner surface into which the grooves (2) have been introduced, in a cross section at right angles to the longitudinal axis (A), has a diameter Di and a radius r.sub.1=Di/2, the grooves (2) in the cross section at right angles to the longitudinal axis (A), in their groove base (4), each have the form of a circular arc and the circular arc has a radius r.sub.2, the grooves (2) each have a groove depth TT which, in the cross section at right angles to the longitudinal axis (A), corresponds in each case to the smallest distance between the circle having the diameter Di on which the inner surface lies and the center of which lies on the longitudinal axis (A), and the furthest removed point of the groove base (4) of the groove (2) from the longitudinal axis (A), wherein the numerical value |D.sub.eqv| of an equivalent diameter D.sub.eqv and the number N.sub.T of grooves (2) and the numerical value |TT| of the groove depth TT of the grooves (2) measured in mm satisfy the relationship
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=C1+C2*|TT|+C3*VD+C4*|D.sub.eqv|+(|TT|−C5)*(VD−C6)*C7+(|TT|−C5)*(|D.sub.eqv|−C8)*C9 with the constants C1=1946.066 C2=302.378 C3=−2.178 C4=266.002 C5=1.954 C6=50.495 C7=−2.004 C8=79.732 C9=−1.041 C10=0.04631 C11=−0.26550 −0.2≥P1≥−0.3 310≤P2≤315 200≤P3≤1500 where the groove density VD that describes the ratio of the grooves N.sub.T in the pipe in relation to the reference number N.sub.ref of the maximum number of grooves having a groove depth TT=1.3 mm that can be introduced in the inner surface area of a pipe having the same equivalent diameter D.sub.eqv, in percent is found from the following relationship:
VD=N.sub.T/N.sub.ref*100 and the reference number N.sub.ref is the greatest natural number that satisfies the relationship $N_{\mathrm{ref}}\le \frac{\pi }{\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{\mathrm{eqv}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{\mathrm{eqv}}.Math.}\right)}\mathrm{where}$ $.Math.r_{\mathrm{eqv}}.Math.=.Math.\sqrt{\frac{.Math.A_{\mathrm{eqv}}.Math.}{\pi }}.Math.$ $A_{\mathrm{eqv}}=A_1+N_T.Math.A_T$ $A_1=\pi .Math.{.Math.r_1.Math.}^2{A}_T=\left[.Math.r_2.Math..Math.\frac{b_2}{2}-\frac{s.Math.\left(.Math.r_2.Math.-\left(.Math.\mathrm{TT}.Math.+h\right)\right)}{2}\right]-\left[.Math.r_1.Math..Math.\frac{b_1}{2}-\frac{s.Math.\left(.Math.r_1.Math.-h\right)}{2}\right]b_1=2.Math..Math.r_1.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_1.Math.})b_2=2.Math..Math.r_2.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_2.Math.})s=2.Math..Math.\sqrt{2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-{\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2}.Math.h=\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}$ and for which there is an r.sub.Nref which, with reference to the value of A.sub.eqv ascertained by the above relationship, satisfies the following conditions that A.sub.eqv is likewise $A_{\mathrm{eqv}}=\pi .Math.{.Math.r_{N_{\mathrm{ref}}}.Math.}^2+N_{\mathrm{ref}}.Math.[\left[{.Math.r_2.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math..Math.\left(.Math.r_2.Math.-\left(1.3+\frac{2.Math..Math.r_2.Math..Math.1.3-{1.3}^2}{2.Math.\left(.Math.r_{N_{\mathrm{ref}}}.Math.-.Math.r_2.Math.+1.3\right)}\right)\right)\right]-[{.Math.r_{N_{\mathrm{ref}}}.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.$ without infringing the boundary conditions $\pi \ge N_{\mathrm{ref}}.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{N_{\mathrm{ref}}}.Math.}\right),r_{N_{\mathrm{ref}}}<r_{\mathrm{eqv}}$ and where the equivalent diameter D.sub.eqv is found from the relationship D.sub.eqv=2 r.sub.eqv.

2. A pipe for thermal cracking of hydrocarbons in the presence of steam, in which a feed mixture is guided through externally heated pipes, wherein the pipe (1) extends along a longitudinal axis (A) and has a number N.sub.T of grooves (2) that have been introduced into the inner surface of the pipe (1) and extend in a helix around the longitudinal axis (A) along the inner surface, the inner surface into which the grooves (2) have been introduced, in a cross section at right angles to the longitudinal axis (A), has a diameter Di and a radius r.sub.1=Di/2, the grooves (2) in the cross section at right angles to the longitudinal axis (A), in their groove base (4), each have the form of a circular arc and the circular arc has a radius r.sub.2, the grooves (2) each have a groove depth TT which, in the cross section at right angles to the longitudinal axis (A), corresponds in each case to the smallest distance between the circle having the diameter Di on which the inner surface lies and the center of which lies on the longitudinal axis (A), and the furthest removed point of the groove base (4) of the groove (2) from the longitudinal axis (A), wherein the numerical value |D.sub.eqv| of an equivalent diameter D.sub.eqv and the number N.sub.T of grooves (2) and the numerical value |TT| of the groove depth TT of the grooves (2) measured in mm satisfy the relationship
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=C1+C2*|TT|+C3*VD+C4*|D.sub.eqv|+(|TT|−C5)*(VD−C6)*C7+(|TT|−C5)*(|D.sub.eqv|−C8)*C9+(VD−C6)*(|D.sub.eqv|−C8)*C10+(|D.sub.eqv|−C8)*(|D.sub.eqv|−C8)*C11 with the constants C1=1946.066 C2=302.378 C3=−2.178 C4=266.002 C5=1.954 C6=50.495 C7=−2.004 C8=79.732 C9=−1.041 010=0.04631 C11=−0.26550 −0.2≥P1≥−0.3 310≤P2≤315 200≤P3≤1500, where the groove density VD that describes the ratio of the grooves N.sub.T in the pipe in relation to the reference number N.sub.ref of the maximum number of grooves having a groove depth TT=1.3 mm that can be introduced in the inner surface area of a pipe having the same equivalent diameter D.sub.eqv, in percent is found from the following relationship:
VD=N.sub.T/N.sub.ref*100 and the reference number N.sub.ref is the greatest natural number that satisfies the relationship $N_{\mathrm{ref}}\le \frac{\pi }{\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{\mathrm{eqv}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{\mathrm{eqv}}.Math.}\right)}\mathrm{where}$ $.Math.r_{\mathrm{eqv}}.Math.=.Math.\sqrt{\frac{.Math.A_{\mathrm{eqv}}.Math.}{\pi }}.Math.$ $A_{\mathrm{eqv}}=A_1+N_T.Math.A_T$ $A_1=\pi .Math.{.Math.r_1.Math.}^2{A}_T=\left[.Math.r_2.Math..Math.\frac{b_2}{2}-\frac{s.Math.\left(.Math.r_2.Math.-\left(.Math.\mathrm{TT}.Math.+h\right)\right)}{2}\right]-\left[.Math.r_1.Math..Math.\frac{b_1}{2}-\frac{s.Math.\left(.Math.r_1.Math.-h\right)}{2}\right]b_1=2.Math..Math.r_1.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_1.Math.})b_2=2.Math..Math.r_2.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_2.Math.})s=2.Math..Math.\sqrt{2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-{\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2}.Math.h=\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}$ and for which there is an r.sub.Nref which, with reference to the value of A.sub.eqv ascertained by the above relationship, satisfies the following conditions that A.sub.eqv is likewise $A_{\mathrm{eqv}}=\pi .Math.{.Math.r_{N_{\mathrm{ref}}}.Math.}^2+N_{\mathrm{ref}}.Math.[\left[{.Math.r_2.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math..Math.\left(.Math.r_2.Math.-\left(1.3+\frac{2.Math..Math.r_2.Math..Math.1.3-{1.3}^2}{2.Math.\left(.Math.r_{N_{\mathrm{ref}}}.Math.-.Math.r_2.Math.+1.3\right)}\right)\right)\right]-[{.Math.r_{N_{\mathrm{ref}}}.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.$ without infringing the boundary conditions $\pi \ge N_{\mathrm{ref}}.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{N_{\mathrm{ref}}}.Math.}\right),r_{N_{\mathrm{ref}}}<r_{\mathrm{eqv}}$ and where the equivalent diameter D.sub.eqv, is found from the relationship D.sub.eqv=2 r.sub.eqv.

3. The pipe as claimed in claim 1, wherein the inner surface of the pipe is cylindrical and the grooves are introduced into this cylindrical inner surface in such a way that portions of the inner surface that form a cylinder remain between the grooves.

4. The pipe as claimed in claim 1, wherein, in a cross section at right angles to the longitudinal axis, the circular arc segment in the inner surface circle occupied by a portion of the inner surface arranged between two grooves is greater than 1% of the circular arc segment in the inner surface circle occupied by the groove opening of at least one of the grooves adjoining this portion of the inner surface area.

5. The pipe as claimed in claim 1, wherein the diameter Di of the inner surface into which the grooves (2) have been introduced is within a range from 15 mm to 280 mm.

6. The pipe as claimed in claim 1, wherein the groove depth TT is within a range from 0.1 mm to 10 mm.

7. The pipe as claimed in claim 1, wherein the number N.sub.T of grooves (2) results in a groove density within a range from 1% to 347%.

8. The pipe as claimed in claim 1, wherein the grooves (2) run at an angle of 20° to 40°, preferably of 22.5° to 32.5°, based on the longitudinal axis (A).

9. The pipe as claimed in claim 1, wherein the pipe is a centrifugally cast pipe or has been produced from a centrifugally cast pipe by introducing grooves into a centrifugally cast pipe.

10. The pipe as claimed in claim 1, wherein the pipe includes a nickel-chromium-iron alloy having high oxidation and carburization resistance, rupture resistance and creep resistance, composed of 0.05% to 0.6% carbon 20% to 50% chromium 5% to 40% iron 2% to 6% aluminum up to 2% silicon up to 2% manganese up to 1.5% niobium up to 1.5% tantalum up to 6.0% tungsten up to 1.0% titanium up to 1.0% zirconium up to 0.5% yttrium up to 0.5% cerium up to 0.5% molybdenum up to 0.1% nitrogen balance: nickel including melting-related impurities, and especially consists of such an alloy.

11. An apparatus for thermal cracking of hydrocarbons in the presence of steam, in which a feed mixture is guided through externally heated pipes, wherein for at least one of said pipes, the pipe (1) extends along a longitudinal axis (A) and has a number N.sub.T of grooves (2) that have been introduced into the inner surface of the pipe (1) and extend in a helix around the longitudinal axis (A) along the inner surface, the inner surface into which the grooves (2) have been introduced, in a cross section at right angles to the longitudinal axis (A), has a diameter Di and a radius r.sub.1=Di/2, the grooves (2) in the cross section at right angles to the longitudinal axis (A), in their groove base (4), each have the form of a circular arc and the circular arc has a radius r.sub.2, the grooves (2) each have a groove depth TT which, in the cross section at right angles to the longitudinal axis (A), corresponds in each case to the smallest distance between the circle having the diameter Di on which the inner surface lies and the center of which lies on the longitudinal axis (A), and the furthest removed point of the groove base (4) of the groove (2) from the longitudinal axis (A), wherein the numerical value |D.sub.eqv| of an equivalent D.sub.eqv and the number N.sub.T of grooves (2) and the numerical value |TT| of the groove depth TT of the grooves (2) measured in mm satisfy the relationship
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=C1+C2*|TT|+C3*VD+C4*|D.sub.eqv|+(|TT|−C5)*(VD−C6)*C7+(|TT|−C5)*(|D.sub.eqv|−C8)*C9 with the constants C1=1946.066 C2=302.378 C3=−2.178 C4=266.002 C5=1.954 C6=50.495 C7=−2.004 C8=79.732 C9=−1.041 C10=0.04631 C11=−0.26550 −0.2≥P1≥−0.3 310≤P2≤315 200≤P3≤1500, where the groove density VD that describes the ratio of the grooves N.sub.T in the pipe in relation to the reference number N.sub.ref of the maximum number of grooves having a groove depth TT=1.3 mm that can be introduced in the inner surface area of a pipe having the same equivalent diameter D.sub.eqv in percent is found from the following relationship:
VD=N.sub.T/N.sub.ref*100 and the reference number N.sub.ref is the greatest natural number that satisfies the relationship $N_{\mathrm{ref}}\le \frac{\pi }{\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{\mathrm{eqv}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{\mathrm{eqv}}.Math.}\right)}\mathrm{where}$ $.Math.r_{\mathrm{eqv}}.Math.=.Math.\sqrt{\frac{.Math.A_{\mathrm{eqv}}.Math.}{\pi }}.Math.$ $A_{\mathrm{eqv}}=A_1+N_T.Math.A_T$ $A_1=\pi .Math.{.Math.r_1.Math.}^2{A}_T=\left[.Math.r_2.Math..Math.\frac{b_2}{2}-\frac{s.Math.\left(.Math.r_2.Math.-\left(.Math.\mathrm{TT}.Math.+h\right)\right)}{2}\right]-\left[.Math.r_1.Math..Math.\frac{b_1}{2}-\frac{s.Math.\left(.Math.r_1.Math.-h\right)}{2}\right]b_1=2.Math..Math.r_1.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_1.Math.})b_2=2.Math..Math.r_2.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_2.Math.})s=2.Math..Math.\sqrt{2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-{\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2}.Math.h=\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}$ and for which there is an r.sub.Nref which with reference to the value of A.sub.eqv ascertained by the above relationship, satisfies the following conditions that A.sub.eqv is likewise $A_{\mathrm{eqv}}=\pi .Math.{.Math.r_{N_{\mathrm{ref}}}.Math.}^2+N_{\mathrm{ref}}.Math.[\left[{.Math.r_2.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math..Math.\left(.Math.r_2.Math.-\left(1.3+\frac{2.Math..Math.r_2.Math..Math.1.3-{1.3}^2}{2.Math.\left(.Math.r_{N_{\mathrm{ref}}}.Math.-.Math.r_2.Math.+1.3\right)}\right)\right)\right]-[{.Math.r_{N_{\mathrm{ref}}}.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.(=\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.$ without infringing the boundary conditions $\pi \ge N_{\mathrm{ref}}.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{N_{\mathrm{ref}}}.Math.}\right),r_{N_{\mathrm{ref}}}<r_{\mathrm{eqv}}$ and where the equivalent diameter D.sub.eqv is found from the relationship D.sub.eqv=2 r.sub.eqv.

12. A method comprising: thermal cracking of hydrocarbons in the presence of steam by guiding a feed mixture through externally heated pipes configured such that for each said pipe, the pipe (1) extends along a longitudinal axis (A) and has a number N.sub.T of grooves (2) that have been introduced into the inner surface of the pipe (1) and extend in a helix around the longitudinal axis (A) along the inner surface, the inner surface into which the grooves (2) have been introduced, in a cross section at right angles to the longitudinal axis (A), has a diameter Di and a radius r.sub.1=Di/2, the grooves (2) in the cross section at right angles to the longitudinal axis (A), in their groove base (4), each have the form of a circular arc and the circular arc has a radius r.sub.2, the grooves (2) each have a groove depth TT which, in the cross section at right angles to the longitudinal axis (A), corresponds in each case to the smallest distance between the circle having the diameter Di on which the inner surface lies and the center of which lies on the longitudinal axis (A), and the furthest removed point of the groove base (4) of the groove (2) from the longitudinal axis (A), wherein the numerical value IDI of an equivalent diameter D and the number N.sub.T of grooves (2) and the numerical value |TT| of the groove depth TT of the grooves (2) measured in mm satisfy the relationship
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=C1+C2*|TT|+C3*VD+C4*|D.sub.eqv|+(|TT|−C5)*(VD−C6)*C7+(|TT|−C5)*(|D.sub.eqv|−C8)*C9 with the constants C1=1946.066 C2=302.378 C3=−2.178 C4=266.002 C5=1.954 C6=50.495 C7=−2.004 C8=79.732 C9=−1.041 C10=0.04631 C11=−0.26550 −0.2≥P1≥−0.3 310≤P2≤315 200≤P3≤1500, where the groove density VD that describes the ratio of the grooves N.sub.T in the pipe in relation to the reference number N.sub.ref of the maximum number of grooves having a groove depth TT=1.3 mm that can be introduced in the inner surface area of a pipe having the same equivalent diameter D.sub.eqv in percent is found from the following relationship:
VD=N.sub.T/N.sub.ref*100 and the reference number N.sub.ref is the greatest natural number that satisfies the relationship $N_{\mathrm{ref}}\le \frac{\pi }{\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{\mathrm{eqv}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{\mathrm{eqv}}.Math.}\right)}\mathrm{where}$ $.Math.r_{\mathrm{eqv}}.Math.=.Math.\sqrt{\frac{.Math.A_{\mathrm{eqv}}.Math.}{\pi }}.Math.$ $A_{\mathrm{eqv}}=A_1+N_T.Math.A_T$ $A_1=\pi .Math.{.Math.r_1.Math.}^2{A}_T=\left[.Math.r_2.Math..Math.\frac{b_2}{2}-\frac{s.Math.\left(.Math.r_2.Math.-\left(.Math.\mathrm{TT}.Math.+h\right)\right)}{2}\right]-\left[.Math.r_1.Math..Math.\frac{b_1}{2}-\frac{s.Math.\left(.Math.r_1.Math.-h\right)}{2}\right]b_1=2.Math..Math.r_1.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_1.Math.})b_2=2.Math..Math.r_2.Math..Math.\arcsin (\frac{.Math.\sqrt{\begin{array}{c}2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-\\ {\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2\end{array}}.Math.}{.Math.r_2.Math.})s=2.Math..Math.\sqrt{2.Math..Math.r_1.Math..Math.\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}-{\left(\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}\right)}^2}.Math.h=\frac{2.Math..Math.r_2.Math..Math..Math.\mathrm{TT}.Math.-{.Math.\mathrm{TT}.Math.}^2}{2.Math.\left(.Math.r_1.Math.-.Math.r_2.Math.+.Math.\mathrm{TT}.Math.\right)}$ and for which there is an r.sub.Nref which with reference to the value of A.sub.eqv ascertained b the above relationship, satisfies the following conditions that A.sub.eqv is likewise $A_{\mathrm{eqv}}=\pi .Math.{.Math.r_{N_{\mathrm{ref}}}.Math.}^2+N_{\mathrm{ref}}.Math.[\left[{.Math.r_2.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math..Math.\left(.Math.r_2.Math.-\left(1.3+\frac{2.Math..Math.r_2.Math..Math.1.3-{1.3}^2}{2.Math.\left(.Math.r_{N_{\mathrm{ref}}}.Math.-.Math.r_2.Math.+1.3\right)}\right)\right)\right]-[{.Math.r_{N_{\mathrm{ref}}}.Math.}^2.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_2.Math.}\right)-.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.$ without infringing the boundary conditions $\pi \ge N_{\mathrm{ref}}.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{N_{\mathrm{ref}}}.Math.}\right),r_{N_{\mathrm{ref}}}<r_{\mathrm{eqv}}$ and where the equivalent diameter D.sub.eqv is found from the relationship D.sub.eqv=2 r.sub.eqv.

Description

(1) The invention is elucidated in detail by a drawing that shows merely embodiments of the invention. The figures show:

(2) FIG. 1 a perspective view of a pipe of the invention,

(3) FIG. 2 a first possible cross section of a pipe of the invention in a section plane at right angles to the longitudinal axis of the pipe,

(4) FIG. 3 a second possible cross section of a pipe of the invention in a section plane at right angles to the longitudinal axis of the pipe,

(5) FIG. 4 a diagram that shows, for a pair of numbers N.sub.T of grooves and groove depths TT that leads to good results and for a pair of numbers N.sub.T of grooves and groove depths TT that leads to further-improved results, the dependence of the heat transfer achieved with this pair on the internal diameter and

(6) FIG. 5 a cross section through a pipe of the invention with a groove.

(7) The inventive pipe 1 shown in FIG. 1 extends along a longitudinal axis A and has a number of 3 grooves 2 introduced into the inner surface that extend in a helix around the longitudinal axis A along the inner surface.

(8) In the cross section of the inventive pipe 1 shown in FIG. 2, it is apparent that, in a preferred embodiment, the grooves 2 are introduced into the otherwise cylindrical inner surface of the pipe 1. Between the grooves 2, there thus remain portions of the cylindrical inner surface of the pipe 1.

(9) Included in FIG. 2 is the groove depth TT and the diameter Di and the inner surface circle 3.

(10) It is likewise shown in FIG. 2 that the cross section of the grooves 2 can be represented by a circular arc.

(11) In the cross section of the inventive pipe 1 shown in FIG. 3, it is apparent that, in an alternative embodiment, the concave grooves in the groove base 4 can merge into a convex shape in the direction of the groove opening 5, and that the portion of the inner surface that remains between two grooves 2 shrinks virtually down to a line. Included in FIG. 3 is the groove depth TT and the diameter Di and the inner surface circle 3.

(12) FIG. 4 shows the values of (H.sub.min(D.sub.eqv, TT.sub.min, VD.sub.max) [watts]) and H.sub.Max(D.sub.eqv, TT.sub.max, VD.sub.min) [watts]) that are reported in the table as a function of the equivalent diameter D.sub.eqv. It is apparent that these values can be represented by a line in each case.

(13) FIG. 5 and the detail Y shown in FIG. 5 show, by way of example, in an inventive pipe with a groove, the nomenclature of the abbreviations A.sub.1, r.sub.1, TT, h, b.sub.2, b.sub.1, A.sub.T, r.sub.2 and s used in the claims and this description.

(14) The way in which the four values N.sub.T, Di, r.sub.2 and TT that characterize the pipe may be found can be shown by the examples that follow.

(15) In one example, there is the external requirement that the passage area is to correspond to that of a smooth pipe of diameter 60 mm. In addition, from a manufacturing point of view, the tools usable for the manufacture of the pipe result in the restriction that a groove depth TT of 1.3 mm and a radius r.sub.2 of the circular arc of the groove base of 8 mm is to be chosen in the case of grooves having a cross section in the shape of a circular arc. The question is what diameter Di and what number of grooves can improve the economic viability of the thermal cracking of hydrocarbons in pipe furnaces with externally heated pipes.

(16) The starting point is thus:

(17) D.sub.eqv=60 mm

(18) A.sub.eqv=π(60/2)=2827.43 mm.sup.2

(19) TT=1.3 mm

(20) r.sub.2=8 mm

(21) A.sub.eqv directly gives r.sub.eqv=Di.sub.eqv/2=30 mm

(22) r.sub.2 and r.sub.eqv give, for the determination of the N.sub.ref in the first step by the formula

(23) 0$N_{\mathrm{ref}}\le \frac{\pi }{\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{\mathrm{eqv}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{\mathrm{eqv}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{\mathrm{eqv}}.Math.}\right)}$
a first N.sub.ref of 18. With this N.sub.ref of 18, the above-described Goal Seek function gives an r.sub.Nref of 29.1406241, with which the secondary condition

(24) $\pi \ge N_{\mathrm{ref}}.Math.\arcsin \left(\frac{.Math.\sqrt{2.Math..Math.r_{N_{\mathrm{ref}}}.Math..Math.\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}-{\left(\frac{\begin{array}{c}2.Math..Math.r_2.Math..Math.\\ 1.3-{1.3}^2\end{array}}{2.Math.\begin{array}{c}(.Math.r_{N_{\mathrm{ref}}}.Math.-\\ .Math.r_2.Math.+1.3)\end{array}}\right)}^2}.Math.}{.Math.r_{N_{\mathrm{ref}}}.Math.}\right),r_{N_{\mathrm{ref}}}<r_{\mathrm{eqv}}$
is simultaneously fulfilled. The number 18 should thus be used as N.sub.ref.

(25) N.sub.ref=18 gives VD=N.sub.T/18*100.

(26) Inserting the minimum values of P1, P2 and P3, for the left-hand term of the equation
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=C1+C2*|TT|+C3*VD+C4*|D.sub.eqv|+(|TT|−C5)*(VD−C6)*C7+(|TT|−C5)*(|D.sub.eqv|−C8)*C9+(VD−C6)*(|D.sub.eqv|−C8)*C10+(|D.sub.eqv|−C8)*(|D.sub.eqv|−C8)*C11

(27) with the constants C1=1946.066 C2=302.378 C3=−2.178 C4=266.002 C5=1.954 C6=50.495 C7=−2.004 C8=79.732 C9=−1.041 C10=0.04631 C11=−0.26550 −0.2≥P1≥−0.3 310≤P2≤315 200≤P3≤1500

(28) gives the value
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=−0.3*(60).sup.2+310*60+200=17720

(29) and inserting the maximum values of P1, P2 and P3 gives, for the left-hand term of the equation,
P1*|D.sub.eqv|.sup.2+P2*|D.sub.eqv|+P3=−0.2*(60).sup.2+315*60+1500=19680

(30) For the right-hand term of the equation
C1+C2*|TT|+C3*VD+C4*|D.sub.eqv|+(|TT|−C5)*(VD−C6)*C7+(|TT|−C5)*(|D.sub.eqv|−C8)*C9+(VD−C6)*(|D.sub.eqv|−C8)*C10+(|D.sub.eqv|−C8)*(|D.sub.eqv|−C8)*C11

(31) inserting |TT|=1.3 and |D.sub.eqv|=60 gives
1946.066+302.378*1.3+−2.178*VD+266.002*60+(1.3−1.954)*(VD−50.495)*−2.004+(1.3−1.954)*(60−79.732)*−1.041+(VD−50.495)*(60−79.732)*0.04631+(60−79.732)*(60−79.732)*−0.26550
and so:
18162.4329−1.7812VD
and with VD=N.sub.T/N.sub.ref*100=N.sub.T/18*100=5.5556 N.sub.T the result is
18162.4329−9.8954N.sub.T

(32) In order to ensure that the pipe achieves the advantages of the invention, N.sub.T should be chosen such that the relationship
19680≥18162.4329−9.8954N.sub.T and the relationship
18162.4329−9.8954N.sub.T≥17720
are fulfilled. Both relationships would be fulfilled with 1≤N.sub.T≤44.71.

(33) Since the N.sub.T thus found is greater than the previously calculated parameter N.sub.ref, even in the case of introduction of the maximum possible number of grooves (N.sub.ref=18), the advantages of the invention can still be achieved at this valley depth. The user is thus at liberty in this working example to endow the pipe with up to the maximum possible number of grooves without losing the advantages of the invention.

(34) The N.sub.T thus found can be used to iteratively determine the radius r.sub.1 of the pipe and hence the internal diameter Di (=2 r.sub.1) of the pipe using the formula (1), since A.sub.eqv=2827.43 mm.sup.2.

(35) Therefore, it is possible to determine all the parameters needed for the manufacture of the pipe that implements the benefits of the invention.