Flow rate distribution and effect of convection and radiation heat transfer on the temperature profile during a coil annealing process

Abstract

Determining the temperature of several steel coils, heated in a furnace with a controlled hydrogen environment is important in an annealing process. Temperatures must be defined during heat treatment in order to guarantee metallurgical properties and acceptable reduced residual stresses. In this paper we approach hydrogen flow characteristics in the furnace and through a set of coils using an annealing non-differential model. Fluid flow is schematized as a pipe network solved by the Hardy Cross method to obtain pressure drops across the various gas flow segments. A comparison is made between measured and simulated results, confirming the adequacy of adopted assumptions and the validity of proposed model. Convective and radiative exchanges between the furnace and the coils are calculated by a discretization using the finite differences method. The convection coefficients are estimated and introduced into the boundary conditions around the coil to obtain the temperature distribution in the coils and in the covering bell. Finally, heat exchanges by convection and radiation are estimated by this model and the modeling errors are <8 °C.

Appendices

Appendix 1: Pressure drop coefficients

Using books and mementos [9–12], expressions of the pressure drop coefficients on all the singularities encountered in the annealing base are obtained.

• Gas entering in the pipe: χ = 0.5

• Gas outlet in the pipe: χ = 1.06

• Singular pressure drop upper coil: χ = 1

• Pressure drop intrados: χ = 1

• Singular pressure drop at the top of the bell: χ = 2

Singular pressure drop stack top hat: the convector on top of stack, a hole was made to change the distribution of flows in base. Laboratory tests have estimated the pressure drops based on the ratio $\frac{{d_{hp} }}{{D_{hp} }}$ (Fig. 15). We modeled these losses by the regression formula

Fig. 15

Schema of the top coil convector under the bell

$$ \chi = \frac{2.94506}{{\left( {{{d_{hp} } \mathord{\left/ {\vphantom {{d_{hp} } D}} \right. \kern-0pt} D}_{hp} } \right)^{2} }} - \frac{0.47641}{{\left( {{{d_{hp} } \mathord{\left/ {\vphantom {{d_{hp} } D}} \right. \kern-0pt} D}_{hp} } \right)}}\quad [9] $$

We introduce this formula into the calculation of pressure drops because the diameter of the orifice top of the stack d _hp , although the fixed construction; could be variable in order to better distribute the flow in the database.

Currently $\frac{{d_{hp} }}{{D_{hp} }} = \frac{100}{605} = 0.165,$ gives a pressure drop coefficient χ = 105.

Appendix 2: Gas temperature calculation

The case of an annealing of 3 coils is taken into account (Fig. 10). The calculation of the temperatures will be carried out according to the 4 steps:

Step 1

Knowing the temperature in A $(\theta_{g} ),$ one will calculate by Eq. (14) the temperature in the branch $\overline{AB}$. The temperature in B, $\theta_{B}$ is known.

Step 2

Knowing the temperature in A $(\theta_{g} ),$ one will calculate by Eq. (14) the temperature in the branch $\overline{AC}$, and then $\theta_{C}$ is known.

Step 3

Knowing the temperature at C, one will calculate by Eq. (17) the temperature in the branch $\overline{CK}$, and then the temperature $\theta_{K}$ in the branch is known. This same reasoning is continued to calculate sequentially $\theta_{D} , \theta_{H} , \theta_{E} , \theta_{F}$ by Eq. (19) and $\theta_{G} , \theta_{N}$ by Eq. (23).

Step 4

Knowing $\theta_{C}$ and $\theta_{H}$, by Eq. (25), one will calculate $\theta_{I}$. By a similar reasoning one will calculate $\theta_{I} ,\theta_{L}$ and $\theta_{0} .$ If the ventilator block is at a thermal balance (low mass), then the temperature in O is identical to the temperature in P (the gas circulates in closed loop and the ventilator does not modify its temperature). The temperature in B is not combined with the temperature of gas flow in P. The 4 steps make it possible to calculate the temperature in P (exit ventilator diffuser) then, by propagation of heat along the gas circuit, the temperature in O (entry ventilator diffuser). While supposing known: convection coefficients; temperatures of the walls (roughly, temperatures of the coils at the step (t − Δt) resulting from the model of propagation of heat in a coil); velocities of the fluid (determined by the model of circulation of gas fluid at the step (t − Δt); the bell temperature (known as a result of the temperature of the gas cane gas and at the temperature of the bell next the gas cane). Thus the temperature in B (exit ventilator diffuser) is identical to the temperature in P (entry ventilator diffuser). If $\theta_{B} \ne \theta_{P} ,$ then at least one of the parameters which models the thermal transfer is incorrect or unknown. By supposing the correct parameters, the only unknown parameter, but fixed in the algorithm at an arbitrary value, is $\theta_{0}^{b}$.

Three cases arise:

Case 1

$\theta_{P} > \theta_{B} ,$ the gas was heated too much by licking the bell and thus the temperature of bell is too high. At the moment t of the calculation, the temperature of the gas cane is constant: according to Eq. (30), it is necessary to decrease $\theta_{0}^{x} ,$ and therefore $\theta_{0}^{b}$ too.

Case 2

$\theta_{B} > \theta_{P } ,$ the gas was not heated enough by licking the bell. Thus the temperature of bell is not high enough. It is necessary to increase $\theta_{0}^{b}$.

Case 3

$\theta_{B} = \theta_{P } ,$ the gas was heated sufficiently on the wall of the bell to guarantee the continuity of the temperatures along the gas circuit: $\theta_{0}^{b}$ is thus well adapted. It is a question of finding in such way that the difference between $\theta_{B}$ and $\theta_{P}$ is nil. Mathematically, the calculation algorithm of the temperature of gas can result in a real scalar function F which at $\theta_{0}^{b}$ forward the value of $(\theta_{B} - \theta_{P} )$.