International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.2.27
[Vol-4, Issue-2, Feb- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
Computational Investigation of Furnace Wall for Silica Ramming Mass with FDM Prof. Nirajkumar C Mehta1, Dr. Dipesh D Shukla2 1
Ph. D. Student, Rai University, Ahmedabad, Gujarat, India 2 Professor, University of Maldives, Maldives
Abstract— Furnaces are useful for melting different materials for casting process. In this research paper, we had done advanced heat transfer analysis of induction furnace wall made of silica ramming mass using explicit finite difference method. We have divided actual geometry of furnace refractory wall into 14 elements and 24 nodes. We have derived explicit finite difference equations for all 24 nodes. We have calculated temperature distribution and thermal stress distribution for all different nodes with respect to time. We have plotted graphs for maximum temperature v/s time and maximum stress v/s time. We found that results indicate the effect of thermal fatigue in the induction furnace wall for silica ramming mass. The analysis is very helpful in understanding how thermal fatigue failure of refractory wall happens. Keywords— Advanced heat transfer analysis, Temperature distribution, Stress distribution, Explicit finite difference method, Silica ramming mass. I. INRODUCTION Furnace is a term used to identify a closed space here heat is applied to a body in order to raise its temperature. The source of heat may be fuel or electricity. Commonly, metals and alloys and sometimes non-metals are heated in furnaces. The purpose of heating defines the temperature of heating and heating rate. Increase in temperature softens the metals. They become amenable to deformation. This softening occurs with or without a change in the metallic structure. Heating to lower temperatures (below the critical temperature) of the metal softens it by relieving the internal stresses. On the other hand, metals heated to temperatures above the critical temperatures leads to changes in crystal structures and recrystallization like annealing. Further some metals and alloys are melted, ceramic products vitrified, coals coked, metals like zinc are vaporized and many other processes are performed in Furnaces. Induction furnaces are widely used in the iron industry for the casting of the different grades of cast iron products. Refractory wall of induction melting furnace is a key www.ijaers.com
component which is used as insulation layer. It is made of ramming mass like silica, silica, magnesia etc. The refractory wall is directly influenced by the thermal cycling of the high temperature molten iron in the furnace. Thermal fatigue failure is easy to happen for it because of the larger phase transformation thermal stresses and it has a shorter life. This can cause serious production accidents. Therefore, the service life problem of the refractory wall has always been a focus of attention in the application of this to the industry. [13-15] The research on the distribution rule of temperature and thermal stress field and on the fatigue life assessment method for the refractory wall will not only lay foundation for the study on the thermal fatigue of this kind of parts under thermal shock condition of low cycle and high phase transition stresses but also offers effective control for thermal fatigue failure. Computational heat transfer, computational fluid dynamic analysis is done for induction melting furnace, refrigerator condenser, induction heating furnace using different numerical methods like finite volume method and finite element method by different researchers. [1-12] Here, Explicit Finite Difference Method is used to find out temperature and thermal stress variation with respect to time. II.
DEVELOPMENT OF ADVANCED HEAT TRANSFER MODEL We have divided Induction Furnace Wall into a Nodal Network as shown in Fig. 1. It is divided into 24 nodes. We have derived Explicit Finite Difference Equations for all nodes as per the boundary conditions applied to it. The furnace wall is having thermal conduction heat transfer between different nodes. It is having atmospheric heat convection ha applied from top side of the furnace wall which is open to atmosphere. It is having heat convection from molten metal from inside which is hi. It is having heat convection ho from cooling water which is circulating outside the furnace wall. [16] Page | 133
International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.2.27 To solve this advanced heat transfer problem of induction melting furnace wall which is made from Silica Ramming Mass, the following initial and boundary conditions, material properties and basic assumptions are made: ďƒ˜ Refractory Materials for induction furnace wall meets the basic assumptions in the science of mechanics. ďƒ˜ Environmental Temperature is homogeneous at 27° C. ďƒ˜ Ignore the influence of heat radiation. ďƒ˜ Ignore the effect of gravity field. ďƒ˜ The surface of induction melting furnace wall is clean. ďƒ˜ The initial temperature of the induction melting furnace is set 27° C and it is agreement with the ambient temperature during solving the problem. ďƒ˜ Heat convections are considered constant for this analysis. ďƒ˜ Scarp material input inside furnace is considered uniform for our analysis.
[Vol-4, Issue-2, Feb- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
Node 1: â„Žđ?‘Ž đ?œŒ
∆đ?‘Ľ 2
∆đ?‘Ľ ∆đ?‘Ś
âˆ
2 2
�1�+1 k
(đ?‘‡âˆž − đ?‘‡1đ?‘– )+ â„Žđ?‘œ
2
(đ?‘‡âˆž − đ?‘‡1đ?‘– ) + k
∆đ?‘Ś đ?‘‡2đ?‘– − đ?‘‡1đ?‘– 2
∆đ?‘Ľ
∆đ?‘Ľ đ?‘‡4đ?‘– − đ?‘‡1đ?‘–
+k
2
∆đ?‘Ś
=
đ?‘‡1đ?‘–+1 −đ?‘‡1đ?‘– ∆đ?‘Ą
= ((â„Žđ?‘Ž
∆đ?‘Ľ đ?‘‡4đ?‘– − đ?‘‡1đ?‘– 2
∆đ?‘Ś
∆đ?‘Ś
)
∆đ?‘Ľ
− đ?‘‡1đ?‘– )+ â„Žđ?‘œ
(đ?‘‡ 2 ∞ 4∆đ?‘Ą
đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
∆đ?‘Ś 2
(đ?‘‡âˆž − đ?‘‡1đ?‘– ) + k
∆đ?‘Ś đ?‘‡2đ?‘– − đ?‘‡1đ?‘– 2
∆đ?‘Ľ
+
)+ �1�
T[1][i+1] = (((0.5*ha*x*(To-T[1][i])) +(ho*y*(TaT[1][i])/2)+(0.5*k*y*(T[2][i]T[1][i])/x)+(0.5*k*x*(T[4][i]-T[1][i])/y) )*((4*t)/(r*c*x*y))) +T[1][i]; Node 2: ∆đ?‘Ś đ?‘‡1đ?‘– − đ?‘‡2đ?‘–
â„Žđ?‘Žâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡2đ?‘– )+ k ∆đ?‘Ś
đ?œŒâˆ†đ?‘Ľ
2
∆đ?‘Ľ
+k
∆đ?‘Ś đ?‘‡3đ?‘– − đ?‘‡2đ?‘– 2
∆đ?‘Ľ
+ k∆đ?‘Ľ
đ?‘‡5đ?‘– − đ?‘‡2đ?‘–
=
∆đ?‘Ś
đ?‘‡2đ?‘–+1 −đ?‘‡2đ?‘–
âˆ
∆đ?‘Ą ∆đ?‘Ś đ?‘‡1đ?‘– − đ?‘‡2đ?‘–
đ?‘‡2đ?‘–+1 k∆đ?‘Ľ
2
= ((â„Žđ?‘Žâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡2đ?‘– )+ k
đ?‘‡5đ?‘– − đ?‘‡2đ?‘– ∆đ?‘Ś
2∆đ?‘Ą
)
đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
2
∆đ?‘Ľ
+k
∆đ?‘Ś đ?‘‡3đ?‘– − đ?‘‡2đ?‘– 2
∆đ?‘Ľ
+
) + �2�
T[2][i+1] = (((ha*x*(To-T[2][i]))+(0.5*k*y*(T[1][i]T[2][i])/x)+(0.5*k*y*(T[3][i]-T[2][i])/x) +(0.5*k*x*(T[5][i]-T[2][i])/y))*((2*t)/(r*c*x*y))) +T[2][i];
Node 3: â„Žđ?‘Ž đ?œŒ
∆đ?‘Ľ 2
(đ?‘‡âˆž − đ?‘‡3đ?‘– )+ â„Žđ?‘–
∆đ?‘Ľ ∆đ?‘Ś 2 2
�3�+1 = k
âˆ
2
(đ?‘‡â„Ž − đ?‘‡3đ?‘– ) + k
∆đ?‘Ś đ?‘‡2đ?‘– − đ?‘‡3đ?‘– 2
∆đ?‘Ľ
∆đ?‘Ľ đ?‘‡6đ?‘– − đ?‘‡3đ?‘–
+k
2
∆đ?‘Ś
=
đ?‘‡3đ?‘–+1 −đ?‘‡3đ?‘– ∆đ?‘Ą ∆đ?‘Ľ
(( â„Žđ?‘Ž
∆đ?‘Ľ đ?‘‡6đ?‘– − đ?‘‡3đ?‘– 2
∆đ?‘Ś
∆đ?‘Ś
)
(đ?‘‡âˆž − đ?‘‡3đ?‘– )+ â„Žđ?‘–
2 4∆đ?‘Ą
đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
∆đ?‘Ś 2
∆đ?‘Ś đ?‘‡2đ?‘– − đ?‘‡3đ?‘–
(đ?‘‡â„Ž − đ?‘‡3đ?‘– ) + k
2
∆đ?‘Ľ
+
) + �3�
T[3][i+1] = (((ha*x*(To-T[3][i])*0.5) +( hi*y*(ThT[3][i])*0.5 ) +(0.5*k*y*(T[2][i]T[3][i])/x)+(0.5*k*x*(T[6][i]-T[3][i])/y)) *((4*t)/(r*c*x*y))) +T[3][i]; Node 4: â„Žđ?‘œâˆ†đ?‘Ś(đ?‘‡âˆž − đ?‘‡4đ?‘– ) + k∆đ?‘Ś Fig.1: Nodal network for finite difference method
đ?œŒ
∆đ?‘Ľ 2
∆đ?‘Śâˆ
đ?‘‡5đ?‘– − đ?‘‡4đ?‘– ∆đ?‘Ľ
+k
∆đ?‘Ľ đ?‘‡1đ?‘– − đ?‘‡4đ?‘– 2
k
www.ijaers.com
2
∆đ?‘Ś
2
∆đ?‘Ś
=
đ?‘‡4đ?‘–+4 −đ?‘‡4đ?‘– ∆đ?‘Ą
đ?‘‡4đ?‘–+4 = ((â„Žđ?‘œâˆ†đ?‘Ś(đ?‘‡âˆž − đ?‘‡4đ?‘– ) + k∆đ?‘Ś ∆đ?‘Ľ đ?‘‡7đ?‘– − đ?‘‡4đ?‘–
∆đ?‘Ś
∆đ?‘Ľ đ?‘‡7đ?‘– − đ?‘‡4đ?‘–
+k
)
2∆đ?‘Ą đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
đ?‘‡5đ?‘– − đ?‘‡4đ?‘– ∆đ?‘Ľ
+k
∆đ?‘Ľ đ?‘‡1đ?‘– − đ?‘‡4đ?‘– 2
∆đ?‘Ś
+
) + �4�
Page | 134
International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.2.27 T[4][i+1] = ((( ho*y*(Ta-T[4][i])) +(k*y*(T[5][i]T[4][i])/x)+(0.5*k*x*(T[1][i]-T[4][i])/y) + (0.5*k*x*(T[7][i]-T[4][i])/y)) *((2*t)/(r*c*x*y))) +T[4][i];
[Vol-4, Issue-2, Feb- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
T[8][i+1] = (((k*y*(T[7][i]-T[8][i])/x)+( k*y*(T[9][i]T[8][i])/x ) +(k*x*(T[5][i]-T[8][i])/y) + (k*x*(T[11][i]T[8][i])/y)) *((t)/(r*c*x*y))) +T[8][i];
Node 5: 𝑇4𝑖 − 𝑇5𝑖
k∆𝑦
∆𝑥
𝜌∆𝑥∆𝑦∁ 𝑇5𝑖+1 = k∆𝑥
+ k∆𝑦
𝑇6𝑖 − 𝑇5𝑖 ∆𝑥
+ k∆𝑥
𝑇2𝑖 − 𝑇5𝑖 ∆𝑦
+ k∆𝑥
𝑇8𝑖 − 𝑇5𝑖 ∆𝑦
Node 9:
=
ℎ𝑖∆𝑦(𝑇ℎ − 𝑇9𝑖 ) + k∆𝑦
𝑇5𝑖+1 −𝑇5𝑖 ∆𝑡
((k∆𝑦
𝑇8𝑖 − 𝑇5𝑖 ∆𝑦
𝑇4𝑖 − 𝑇5𝑖
+ k∆𝑦
∆𝑥 ∆𝑡
)
𝑇6𝑖 − 𝑇5𝑖 ∆𝑥
+ k∆𝑥
𝑇2𝑖 − 𝑇5𝑖
𝜌
+
∆𝑦
T[5][i+1] = (((k*y*(T[4][i]-T[5][i])/x)+(k*y*(T[6][i]T[5][i])/x) + (k*x*(T[8][i]-T[5][i])/y)) *((t)/(r*c*x*y))) +T[5][i]; Node 6: ℎ𝑖∆𝑦(𝑇ℎ − 𝑇6𝑖 ) + k∆𝑦 𝜌
∆𝑥 2
∆𝑦∁
𝑇6𝑖+1 = k
2
∆𝑦
∆𝑥
+k
∆𝑥 𝑇3𝑖 − 𝑇6𝑖 2
∆𝑦
∆𝑥 𝑇9𝑖 − 𝑇6𝑖
+k
2
∆𝑦
=
𝑇6𝑖+1 −𝑇6𝑖
)
𝑇6𝑖 )
2∆𝑡 𝜌∁∆𝑥∆𝑦
+ k∆𝑦
𝑇2𝑖 − 𝑇6𝑖 ∆𝑥
+k
∆𝑥 𝑇3𝑖 − 𝑇6𝑖 2
∆𝑦
+
) + 𝑇6𝑖
ℎ𝑜∆𝑦(𝑇∞ − 𝑇7𝑖 ) + k∆𝑦 2
∆𝑦∁
𝑇7𝑖+7 k
∆𝑥
+k
∆𝑥 𝑇4𝑖 − 𝑇7𝑖 2
∆𝑦
𝑖 − 𝑇𝑖 ∆𝑥 𝑇10 7
+k
2
∆𝑦
=
𝑇7𝑖+7 −𝑇7𝑖
= (( ℎ𝑜∆𝑦(𝑇∞ − ∆𝑦
)
2∆𝑡 𝜌∁∆𝑥∆𝑦
𝑇7𝑖 )
+ k∆𝑦
𝑇8𝑖 − 𝑇7𝑖 ∆𝑥
+k
∆𝑥 𝑇4𝑖 − 𝑇7𝑖 2
∆𝑦
+
) + 𝑇7𝑖
2
2
∆𝑦
𝑖 − 𝑇𝑖 ∆𝑥 𝑇12 9
+k
2
∆𝑦
=
∆𝑡
∆𝑦
2∆𝑡
)
𝜌∁∆𝑥∆𝑦
𝑇8𝑖 − 𝑇9𝑖 ∆𝑥
+k
∆𝑥 𝑇6𝑖 − 𝑇9𝑖 2
∆𝑦
+
) + 𝑇9𝑖
T[9][i+1] = (( (hi*y*(Th-T[9][i]))+(k*y*(T[8][i]T[9][i])/x)+ (0.5*k*x*(T[6][i]-T[9][i])/y) + (0.5*k*x*(T[12][i]-T[9][i])/y)) *((2*t)/(r*c*x*y))) +T[9][i]; Node 10:
∆𝑥 2
∆𝑦∁
k
2
𝑖 − 𝑇𝑖 𝑇11 10
∆𝑥
+k
𝑖 ∆𝑥 𝑇7𝑖 − 𝑇10
2
∆𝑦
𝑖 − 𝑇𝑖 ∆𝑥 𝑇15 10
+k
2
∆𝑦
=
𝑖+10 −𝑇 𝑖 𝑇10 10
∆𝑡
𝑖 = ((ℎ𝑜∆𝑦(𝑇∞ − 𝑇10 ) + k∆𝑦
𝑖 − 𝑇𝑖 ∆𝑥 𝑇15 10
∆𝑦
)
2∆𝑡 𝜌∁∆𝑥∆𝑦
𝑖 − 𝑇𝑖 𝑇11 10
∆𝑥
+k
𝑖 ∆𝑥 𝑇7𝑖 − 𝑇10
2
∆𝑦
+
𝑖 ) + 𝑇10
T[10][i+1] = (( (ho*y*(Ta-T[10][i]))+(k*y*(T[11][i]T[10][i])/x)+ (0.5*k*x*(T[7][i]-T[10][i])/y) + (0.5*k*x*(T[15][i]-T[10][i])/y)) *((2*t)/(r*c*x*y))) +T[10][i]; Node 11: 𝑖 − 𝑇𝑖 𝑇10 11
∆𝑥
𝜌∆𝑥∆𝑦∁ 𝑖+1 𝑇11
T[7][i+1] = (((ho*y*(Ta-T[7][i])) + (k*y*(T[8][i]T[7][i])/x)+(0.5*k*x*(T[4][i]-T[7][i])/y) + (0.5*k*x*(T[10][i]-T[7][i])/y)) *((2*t)/(r*c*x*y))) +T[7][i];
∆𝑥 𝑇6𝑖 − 𝑇9𝑖
𝑇9𝑖+1 −𝑇9𝑖
𝑖 − 𝑇𝑖 ∆𝑥 𝑇12 9
k∆𝑦
∆𝑡
𝑖 − 𝑇𝑖 ∆𝑥 𝑇10 7
2
𝑇8𝑖 − 𝑇7𝑖
+k
= ((ℎ𝑖∆𝑦(𝑇ℎ − 𝑇9𝑖 ) + k∆𝑦
𝑖+10 𝑇10
Node 7:
𝜌
k
𝜌
T[6][i+1] = (((hi*y*(Th-T[6][i])) + (k*y*(T[2][i]T[6][i])/x)+(0.5*k*x*(T[3][i]-T[6][i])/y) + (0.5*k*x*(T[9][i]-T[6][i])/y)) *((2*t)/(r*c*x*y))) +T[6][i];
∆𝑥
∆𝑦∁
∆𝑥
𝑖 ℎ𝑜∆𝑦(𝑇∞ − 𝑇10 ) + k∆𝑦
∆𝑡
(( ℎ𝑖∆𝑦(𝑇ℎ −
∆𝑥 𝑇9𝑖 − 𝑇6𝑖
𝑇2𝑖 − 𝑇6𝑖
2
𝑇9𝑖+1
) + 𝑇5𝑖
𝜌∁∆𝑥∆𝑦
∆𝑥
𝑇8𝑖 − 𝑇9𝑖
k∆𝑥
+ k∆𝑦
𝑖 − 𝑇𝑖 𝑇12 11
∆𝑥
+ k∆𝑥
𝑖 𝑇8𝑖 − 𝑇11
∆𝑦
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇16 11
∆𝑦
=
𝑖+1 −𝑇 𝑖 𝑇11 11
∆𝑡
= (( k∆𝑦
𝑖 − 𝑇𝑖 𝑇16 11
∆𝑦
)
𝑖 − 𝑇𝑖 𝑇10 11
∆𝑥 ∆𝑡
𝜌∁∆𝑥∆𝑦
+ k∆𝑦
𝑖 − 𝑇𝑖 𝑇12 11
∆𝑥
+ k∆𝑥
𝑖 𝑇8𝑖 − 𝑇11
∆𝑦
+
𝑖 ) + 𝑇11
T[11][i+1] = (( (k*y*(T[10][i]-T[11][i])/x)+ (k*y*(T[12][i]-T[11][i])/x)+ (k*x*(T[8][i]-T[11][i])/y) + (k*x*(T[16][i]-T[11][i])/y)) *((t)/(r*c*x*y))) +T[11][i];
Node 8: k∆𝑦
𝑇7𝑖 − 𝑇8𝑖 ∆𝑥
𝜌∆𝑥∆𝑦∁
+ k∆𝑦
𝑖 − 𝑇𝑖 𝑇11 8
∆𝑦
∆𝑥
+ k∆𝑥
𝑇5𝑖 − 𝑇8𝑖 ∆𝑦
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇11 8
∆𝑦
𝑇8𝑖+1 −𝑇8𝑖 ∆𝑡
𝑇8𝑖+1 = (( k∆𝑦 k∆𝑥
𝑇9𝑖 − 𝑇8𝑖
)
𝑇7𝑖 − 𝑇8𝑖 ∆𝑥 ∆𝑡
𝜌∁∆𝑥∆𝑦
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+ k∆𝑦
𝑇9𝑖 − 𝑇8𝑖 ∆𝑥
+ k∆𝑥
𝑇5𝑖 − 𝑇8𝑖 ∆𝑦
=
Node 12:
k +
∆𝑦 𝑖 (𝑇ℎ − 𝑇12 ) + ℎ𝑖 (𝑇ℎ 2 2 𝑖 𝑖 𝑖 ∆𝑦 𝑇13 − 𝑇12 ∆𝑥 𝑇9𝑖 − 𝑇12
ℎ𝑖
∆𝑥
2
∆𝑥
+k
2
∆𝑦
𝑖 − 𝑇12 ) + k∆𝑦
+ k∆𝑥
𝑖 𝑖 𝑇17 − 𝑇12
∆𝑦
𝑖 − 𝑇𝑖 𝑇11 12
=𝜌
∆𝑥 3∆𝑥∆𝑦 4
+
∁
𝑖+1 𝑖 𝑇12 −𝑇12
∆𝑡
) + 𝑇8𝑖 Page | 135
International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.2.27 𝑖+1 𝑇12 = (( ℎ𝑖
k
𝑖 − 𝑇𝑖 ∆𝑦 𝑇13 12
2
∆𝑥
∆𝑥 2
𝑖 ∆𝑥 𝑇9𝑖 − 𝑇12
+k
∆𝑦
𝑖 (𝑇ℎ − 𝑇12 ) + ℎ𝑖 2
∆𝑦
+ k∆𝑥
2
𝑖 (𝑇ℎ − 𝑇12 ) + k∆𝑦
𝑖 − 𝑇𝑖 𝑇17 12
∆𝑦
4∆𝑡
)
𝑖 − 𝑇𝑖 𝑇11 12
∆𝑥
+
𝑖 ) + 𝑇12
3𝜌∁∆𝑥∆𝑦
Node 13: 𝑖 − 𝑇𝑖 ∆𝑦 𝑇12 13
𝜌∆𝑥
∆𝑦 2
∁
2
∆𝑥
+k
𝑖 − 𝑇𝑖 ∆𝑦 𝑇14 13
2
∆𝑥
𝑖 − 𝑇𝑖 𝑇18 13
∆𝑦
=
𝑖+1 −𝑇 𝑖 𝑇13 13
𝑖 − 𝑇𝑖 𝑇18 13
∆𝑦
)
2∆𝑡 𝜌∁∆𝑥∆𝑦
2
∆𝑥
+k
𝑖 − 𝑇𝑖 ∆𝑦 𝑇14 13
2
∆𝑥
𝑖 ) + 𝑇13
Node 14: 2
=𝜌
𝑖 (𝑇ℎ − 𝑇14 )+ ℎ𝑖
∆𝑥 ∆𝑦 2 2
∁
k
2
∆𝑦
2
𝑖 (𝑇ℎ − 𝑇14 )+k
𝑖 − 𝑇𝑖 ∆𝑦 𝑇13 14
2
∆𝑥
𝑖 − 𝑇𝑖 ∆𝑥 𝑇19 14
+k
2
∆𝑦
𝑖+1 −𝑇 𝑖 𝑇14 14
)
∆𝑥
𝑖 (𝑇ℎ − 𝑇14 )+ ℎ𝑖
2 4∆𝑡
𝜌∁∆𝑥∆𝑦
∆𝑦 2
𝑖 (𝑇ℎ − 𝑇14 )+k
𝑖 − 𝑇𝑖 ∆𝑦 𝑇13 14
2
∆𝑥
+
T[14][i+1] = (( (0.5*hi*x*(Th-T[14][i]))+(0.5*hi*y*(ThT[14][i])) +(0.5*k*y*(T[13][i]-T[14][i])/x)+ (0.5*k*x*(T[19][i]-T[14][i])/y)) *((4*t)/(r*c*x*y))) +T[14][i]; Node 15: 𝑖 ℎ𝑜∆𝑦(𝑇∞ − 𝑇15 ) + k∆𝑦
𝜌
∆𝑥 2
∆𝑦∁
𝑖 − 𝑇𝑖 𝑇16 15
∆𝑥
+k
𝑖 − 𝑇𝑖 ∆𝑥 𝑇10 15
2
𝑖 − 𝑇𝑖 ∆𝑥 𝑇20 15
2
𝑖 − 𝑇𝑖 ∆𝑥 𝑇20 15
+k
2
∆𝑦
∆𝑡
∆𝑦
)
2∆𝑡 𝜌∁∆𝑥∆𝑦
)+
𝑖 − 𝑇𝑖 𝑇16 15
∆𝑥
+k
𝑖 − 𝑇𝑖 ∆𝑥 𝑇10 15
2
∆𝑦
𝑖 𝑇15
T[15][i+1] = (( (ho*y*(Ta-T[15][i]))+(k*y*(T[16][i]T[15][i])/x)+ (0.5*k*x*(T[10][i]-T[15][i])/y)+( 0.5*k*x*(T[20][i]-T[15][i])/y)) *((2*t)/(r*c*x*y))) +T[15][i]; Node 16:
www.ijaers.com
𝑖 − 𝑇𝑖 𝑇21 16
∆𝑦
=
+ k∆𝑦
∆𝑥
𝜌∁∆𝑥∆𝑦
𝑖 − 𝑇𝑖 𝑇17 16
∆𝑥
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇11 16
∆𝑦
+
𝑖 ) + 𝑇16
k∆𝑥
+ k∆𝑦
∆𝑥
𝑖 − 𝑇𝑖 𝑇18 17
∆𝑥
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇12 17
∆𝑦
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇22 17
∆𝑦
=
𝑖+1 −𝑇 𝑖 𝑇17 17
∆𝑡
= (( k∆𝑦
𝑖 − 𝑇𝑖 𝑇22 17
∆𝑦
𝑖 − 𝑇𝑖 𝑇16 17
∆𝑥 ∆𝑡
)
+ k∆𝑦
𝜌∁∆𝑥∆𝑦
𝑖 − 𝑇𝑖 𝑇18 17
∆𝑥
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇12 17
∆𝑦
+
𝑖 ) + 𝑇17
T[17][i+1] = (( (k*y*(T[16][i]-T[17][i])/x)+ (k*y*(T[18][i]T[17][i])/x )+(k*x*(T[12][i]-T[17][i])/y)+( k*x*(T[22][i]T[17][i])/y)) *((t)/(r*c*x*y))) +T[17][i]; Node 18: 𝑖 − 𝑇𝑖 𝑇17 18
∆𝑥
𝜌∆𝑥∆𝑦∁
k∆𝑥
+ k∆𝑦
𝑖 − 𝑇𝑖 𝑇19 18
∆𝑥
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇13 18
∆𝑦
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇23 18
∆𝑦
=
𝑖+1 −𝑇 𝑖 𝑇18 18
∆𝑡
= ((k∆𝑦
𝑖 − 𝑇𝑖 𝑇23 18
∆𝑦
)
𝑖 − 𝑇𝑖 𝑇17 18
∆𝑥 ∆𝑡
𝜌∁∆𝑥∆𝑦
+ k∆𝑦
𝑖 − 𝑇𝑖 𝑇19 18
∆𝑥
+ k∆𝑥
𝑖 − 𝑇𝑖 𝑇13 18
∆𝑦
+
𝑖 ) + 𝑇18
T[18][i+1] = (( (k*y*(T[17][i]-T[18][i])/x)+ (k*y*(T[19][i]T[18][i])/x )+(k*x*(T[13][i]-T[18][i])/y)+( k*x*(T[23][i]T[18][i])/y)) *((t)/(r*c*x*y))) +T[18][i]; Node 19:
=
𝑖+15 −𝑇 𝑖 𝑇15 15
𝑖+15 𝑖 𝑇15 = (( ℎ𝑜∆𝑦(𝑇∞ − 𝑇15 ) + k∆𝑦
k
∆𝑦
𝑖 − 𝑇𝑖 𝑇15 16
∆𝑡
)
𝑖 − 𝑇𝑖 𝑇16 17
𝑖+1 𝑇18
𝑖 ) + 𝑇14
∆𝑦
+ k∆𝑥
Node 17:
k∆𝑦
∆𝑡
𝑖+1 𝑇14 = (( ℎ𝑖 𝑖 − 𝑇𝑖 ∆𝑥 𝑇19 14
∆𝑦
𝑖 − 𝑇𝑖 𝑇11 16
T[16][i+1] = (( (k*y*(T[15][i]-T[16][i])/x)+ (k*y*(T[17][i]T[16][i])/x )+(k*x*(T[11][i]-T[16][i])/y)+( k*x*(T[21][i]T[16][i])/y)) *((t)/(r*c*x*y))) +T[16][i];
𝑖+1 𝑇17
T[13][i+1] = (( (hi*x*(Th-T[13][i]))+(0.5*k*y*(T[12][i]T[13][i])/x)+ (0.5*k*y*(T[14][i]-T[13][i])/x)+ (k*x*(T[18][i]-T[13][i])/y)) *((2*t)/(r*c*x*y))) +T[13][i];
∆𝑥
∆𝑦
𝜌∆𝑥∆𝑦∁
+
∆𝑥
+ k∆𝑥
∆𝑡
𝑖 − 𝑇𝑖 𝑇21 16
∆𝑡 𝑖 − 𝑇𝑖 ∆𝑦 𝑇12 13
ℎ𝑖
k∆𝑥
𝑖 − 𝑇𝑖 𝑇17 16
𝑖+1 −𝑇 𝑖 𝑇16 16
𝑖+1 𝑇16 = (( k∆𝑦
k∆𝑦
𝑖+1 𝑖 𝑇13 = (( ℎ𝑖∆𝑥(𝑇ℎ − 𝑇13 )+ k
k∆𝑥
+ k∆𝑥
+ k∆𝑦
∆𝑥
𝜌∆𝑥∆𝑦∁
T[12][i+1] = (( (0.5*hi*x*(Th-T[12][i]))+( 0.5*hi*y*(ThT[12][i])) +(k*y*(T[11][i]-T[12][i])/x)+ (0.5*k*y*(T[13][i]-T[12][i])/x) + (0.5*k*x*(T[9][i]T[12][i])/y) + (k*x*(T[17][i]-T[12][i])/y)) *((4*t)/(3*r*c*x*y))) +T[12][i];
𝑖 ℎ𝑖∆𝑥(𝑇ℎ − 𝑇13 )+ k
𝑖 − 𝑇𝑖 𝑇15 16
k∆𝑦
[Vol-4, Issue-2, Feb- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
𝑖 ℎ𝑖∆𝑦(𝑇ℎ − 𝑇19 ) + k∆𝑦
𝜌 +
∆𝑥 2
∆𝑦∁
𝑖+1 𝑇19
k
∆𝑥
+k
𝑖 − 𝑇𝑖 ∆𝑥 𝑇14 19
2
∆𝑦
𝑖 − 𝑇𝑖 ∆𝑥 𝑇24 19
+k
2
∆𝑦
=
𝑖+1 −𝑇 𝑖 𝑇19 19
∆𝑡
𝑖 = ((ℎ𝑖∆𝑦(𝑇ℎ − 𝑇19 ) + k∆𝑦
𝑖 − 𝑇𝑖 ∆𝑥 𝑇24 19
2
𝑖 − 𝑇𝑖 𝑇18 19
∆𝑦
)
2∆𝑡 𝜌∁∆𝑥∆𝑦
𝑖 − 𝑇𝑖 𝑇18 19
∆𝑥
+k
𝑖 − 𝑇𝑖 ∆𝑥 𝑇14 19
2
∆𝑦
+
𝑖 ) + 𝑇19
T[19][i+1] = (( (hi*y*(Th-T[19][i]))+(k*y*(T[18][i]T[19][i])/x)+ (0.5*k*x*(T[14][i]-T[19][i])/y)+( 0.5*k*x*(T[24][i]-T[19][i])/y)) *((2*t)/(r*c*x*y))) +T[19][i]; Node 20:
Page | 136
International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.2.27 â„Žđ?‘œ k
∆đ?‘Ľ 2
đ?‘– (đ?‘‡âˆž − đ?‘‡20 )+ â„Žđ?‘œ
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ľ đ?‘‡15 20
2
∆đ?‘Ś
�+20 �20
k
=
2 2 ∆đ?‘Ľ ((â„Žđ?‘œ (đ?‘‡âˆž 2
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡21 20
2
=đ?œŒ
∆đ?‘Ľ ∆đ?‘Ś
∆đ?‘Ľ
âˆ
∆đ?‘Ś 2
2
−
∆đ?‘Ś
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡21 20
2
∆đ?‘Ľ
â„Žđ?‘œ
+
đ?‘–+20 −đ?‘‡ đ?‘– đ?‘‡20 20
k
∆đ?‘Ą đ?‘– đ?‘‡20 )+
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ľ đ?‘‡15 20
+k
đ?‘– (đ?‘‡âˆž − đ?‘‡20 )+k
)
â„Žđ?‘œ
4∆đ?‘Ą đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
∆đ?‘Ś 2
(đ?‘‡âˆž −
)+
� �20 )
+
∆đ?‘Ľ 2
đ?‘– (đ?‘‡âˆž − đ?‘‡24 )+ â„Žđ?‘–
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ľ đ?‘‡19 24
2
∆đ?‘Ś
=đ?œŒ
đ?‘–+1 đ?‘‡24 = ((â„Žđ?‘œ
� �20
k
T[20][i+1] = (( (0.5*ho*x*(Ta-T[20][i]))+( 0.5*ho*y*(TaT[20][i]))+(0.5*k*y*(T[21][i]-T[20][i])/x)+ (0.5*k*x*(T[15][i]-T[20][i])/y))*((4*t)/(r*c*x*y))) +T[20][i];
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ľ đ?‘‡19 24
2
[Vol-4, Issue-2, Feb- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
∆đ?‘Ś
)
∆đ?‘Ľ
∆đ?‘Ľ ∆đ?‘Ś 2 2
∆đ?‘Ś
âˆ
2
đ?‘– (đ?‘‡â„Ž − đ?‘‡24 )+k
đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
2
∆đ?‘Ľ
+
đ?‘–+1 −đ?‘‡ đ?‘– đ?‘‡24 24
∆đ?‘Ą
đ?‘– (đ?‘‡âˆž − đ?‘‡24 )+ â„Žđ?‘–
2 4∆đ?‘Ą
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡23 24
∆đ?‘Ś 2
đ?‘– (đ?‘‡â„Ž − đ?‘‡24 )+k
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡23 24
2
∆đ?‘Ľ
+
� ) + �24
T[24][i+1] = (( (0.5*ho*x*(Ta-T[24][i]))+( 0.5*hi*y*(ThT[24][i])) +(0.5*k*y*(T[23][i]-T[24][i])/x)+ (0.5*k*x*(T[19][i]-T[24][i])/y))*((4*t)/(r*c*x*y))) +T[24][i];
Node 21: đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡20 21
đ?‘– â„Žđ?‘œâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡21 )+ k ∆đ?‘Ś
đ?œŒâˆ†đ?‘Ľ
2
�+1 �21
k∆đ?‘Ľ
âˆ
2
+k
∆đ?‘Ľ
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡22 21
2
∆đ?‘Ľ
+ k∆đ?‘Ľ
đ?‘– − đ?‘‡đ?‘– đ?‘‡16 21
∆đ?‘Ś
= III. PROGRAMMING & SOLUTION With the help of a computer program we can solve the matrix created by finite difference equations for 24 nodes. We can calculate temperature distribution and stress distribution with respect to time.
đ?‘–+1 −đ?‘‡ đ?‘– đ?‘‡21 21
∆đ?‘Ą đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡20 21
đ?‘– = ((â„Žđ?‘œâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡21 )+ k
đ?‘– − đ?‘‡đ?‘– đ?‘‡16 21
∆đ?‘Ś
)
2∆đ?‘Ą
2
∆đ?‘Ľ
+k
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡22 21
2
∆đ?‘Ľ
+
� ) + �21
đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
T[21][i+1] = (( (0.5*ho*x*(TaT[21][i]))+(0.5*k*y*(T[20][i]-T[21][i])/x)+( 0.5*k*y*(T[22][i]-T[21][i])/x)+ (k*x*(T[16][i]T[21][i])/y))*((2*t)/(r*c*x*y))) +T[21][i]; Node 22: đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡21 22
đ?‘– â„Žđ?‘œâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡22 )+ k ∆đ?‘Ś
đ?œŒâˆ†đ?‘Ľ
2
�+1 �22
k∆đ?‘Ľ
âˆ
2
+k
∆đ?‘Ľ
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡23 22
2
∆đ?‘Ľ
+ k∆đ?‘Ľ
đ?‘– − đ?‘‡đ?‘– đ?‘‡17 22
∆đ?‘Ś
=
đ?‘–+1 −đ?‘‡ đ?‘– đ?‘‡22 22
∆đ?‘Ą đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡21 22
đ?‘– = ((â„Žđ?‘œâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡22 )+ k
đ?‘– − đ?‘‡đ?‘– đ?‘‡17 22
∆đ?‘Ś
)
2∆đ?‘Ą
2
∆đ?‘Ľ
+k
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡23 22
2
∆đ?‘Ľ
+
� ) + �22
đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
T[22][i+1] = (( (ho*x*(Ta-T[22][i]))+(0.5*k*y*(T[21][i]T[22][i])/x)+( 0.5*k*y*(T[23][i]-T[22][i])/x)+ (k*x*(T[17][i]-T[22][i])/y))*((2*t)/(r*c*x*y))) +T[22][i]; Node 23: đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡22 23
đ?‘– â„Žđ?‘œâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡23 )+ k ∆đ?‘Ś
đ?œŒâˆ†đ?‘Ľ
2
�+1 �23
k∆đ?‘Ľ
âˆ
2
+k
∆đ?‘Ľ
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡24 23
2
∆đ?‘Ľ
+ k∆đ?‘Ľ
đ?‘– − đ?‘‡đ?‘– đ?‘‡18 23
∆đ?‘Ś
=
đ?‘–+1 −đ?‘‡ đ?‘– đ?‘‡23 23
∆đ?‘Ą đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡22 23
đ?‘– = ((â„Žđ?‘œâˆ†đ?‘Ľ(đ?‘‡âˆž − đ?‘‡23 )+ k
đ?‘– − đ?‘‡đ?‘– đ?‘‡18 23
∆đ?‘Ś
)
2∆đ?‘Ą đ?œŒâˆ ∆đ?‘Ľâˆ†đ?‘Ś
2
∆đ?‘Ľ
+k
đ?‘– − đ?‘‡đ?‘– ∆đ?‘Ś đ?‘‡24 23
2
∆đ?‘Ľ
+
� ) + �23
T[23][i+1] = (( (ho*x*(Ta-T[23][i]))+(0.5*k*y*(T[22][i]T[23][i])/x)+( 0.5*k*y*(T[24][i]-T[23][i])/x)+ (k*x*(T[18][i]-T[23][i])/y))*((2*t)/(r*c*x*y))) +T[23][i]; Node 24: www.ijaers.com
Table.1: Material Property and Boundary Conditions Material Properties and Boundary Unit Conditions for Silica Ramming Mass 1 Internal Film Co200 W/m2 K efficient hi 2 External Film Co40 W/m2 K efficient ho 3 Atmosphere Film Co10 W/m2 K efficient ha 4 Density 2800 Kg/m3 5 Time Interval Δt 10 Seconds 6 Thermal Conductivity k 1.7 W/m K 7 Temperature outside 303 Kelvin Furnace Wall 8 Temperature inside 1673 Kelvin Furnace Wall 9 Temperature of Air 303 Kelvin 10 Specific Heat 950 J/kg K 11 Elasticity Constant 180000 N/ m2 12 Thermal Expansion Co0.00000 m/ K efficient 122 13 Ultimate Stress 500 MPa IV. RESULTS AND DISCUSSION We can see from the Fig. 2 that maximum temperature is increasing from atmospheric temperature 300 K and reaches to maximum temperature 1623 K in 45 minutes and then starts reducing and reaches to 724 K in next 15 minutes. It again starts increasing and reaches to maximum 1623 K Page | 137
International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.2.27 after 105 minutes and again starts reducing. There are 10 similar temperature cycles in one day. We can see from the Fig. 3 that maximum thermal stress is increasing from initial condition 0 MPa and reaches to maximum stress 356MPa in 45 minutes and then starts reducing and reaches to 159MPa in next 15 minutes. It again starts increasing and reaches to maximum stress 356 MPa after 105 minutes and again it starts reducing. There are 10 similar thermal stress cycles in one day.
wall of silica ramming mass is under the effect of low cycle thermal fatigue loading. The reason for its low life span is low cycle thermal fatigue load.
[1]
[2]
[3]
[4]
1 43 85 127 169 211 253 295 337 379 421 463 505 547 589
Temperature (Kelvin)
Maximum Temperature v/s Time Silica Ramming Mass 1800 1600 1400 1200 1000 800 600 400 200 0
[5]
Time (Minutes)
Fig.2: Maximum Temperature v/s Time Graph for Silica Ramming Mass
Maximum Stress v/s Time Silica Rammimg Mass
[6]
Stress (MPa)
400 300 200
[7]
100
1 43 85 127 169 211 253 295 337 379 421 463 505 547 589
0 Time (Minutes)
[8]
Fig.3: Maximum Stress v/s Time Graph for Silica Ramming Mass V. CONCLUSION We have found deviation of maximum temperature and maximum thermal stress with reference to time for induction furnace wall made up of alumina ramming mass. From the graph, we can conclude that induction furnace
www.ijaers.com
[Vol-4, Issue-2, Feb- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
[9]
REFERENCES Nirajkumar C Mehta, Dipesh D Shukla, Ravi K Popat, "Optimisation of Wall Thickness for Minimum Heat Loss for Induction Furnace by FEA", Indian Foundry Journal, Vol. 60, No. 12, December 2014 Akash D. Raiyani, N. R. Sheth, Niraj C. Mehta, "Thermal Analysis of Hot Wall Condenser for Domestic Refrigerator", International Journal of Science and Research, Volume 3, Issue 7, July 2014 N C Mehta, Vasim G. Machhar, Ravi K. Popat, "Thermal Fatigue Analysis of Induction Furnace Wall for Alumina ramming mass", International Journal of Science and Engineering Applications, Volume 2, Issue 10, 2013 Vimal R. Nakum, Kevin M. Vyas, Niraj C. Mehta, "Research on Induction Heating -A Review", International Journal of Science and Engineering Applications, Volume 2, Issue 6, 2013 Prof. N. C. Mehta , Mr. Akash D. Raiyani , Mr. Vikas R. Gondalia, "Thermal Fatigue Analysis of Induction Melting Furnace Wall for Silica ramming mass", International Journal of Emerging Technology and Advanced Engineering, Volume 3, Issue 2, February 2013 Prof. N. C. Mehta, Mr. Vipul B. Gondaliya, Mr. Jayesh V. Gundaniya, "Applications of Different Numerical Methods in Heat Transfer - A Review", International Journal of Emerging Technology and Advanced Engineering, Volume 3, Issue 2, February 2013 Vipul Gondaliya, Mehul Pujara, Nirajkumar Mehta, "Transient Heat Transfer Analysis of Induction Furnace by Using Finite Element Analysis", International Journal of Applied Research”, Volume 3, Issue 8, August 2013 Nirajkumar C Mehta, Dr. Dipesh D. Shukla, “Thermal Fatigue Analysis of Induction Furnace Wall for Magnesia Ramming Mass”, ASME 2015 Applied Mechanics And Materials Conference, At Seattle, Washington, United States of America, Volume: 12, June 2015 Niraj C Mehta, Dr. Dipesh D. Shukla, Vishvash B. Rajyaguru, “Thermal Fatigue Analysis of Induction Furnace Wall for Zirconia”, National Conference on
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Recent Research and Development in Core Disciplines of Engineering, Vadodara, Volume: 2, April 2015 [10] Nirajkumar C. Mehta, Dr. Dipesh D. Shukla, Vishvash B. Rajyaguru, “Numerical Analysis of Furnace: Review”, National Conference on Recent Research and Development in Core Disciplines of Engineering, Vadodara, Volume: 2, April 2015 [11] Nirajkumar C Mehta, Dr. Dipesh D. Shukla, Pragnesh D. Kandoliya, “Comparison Of Finite Difference Method And Finite Element Method For 2 D Transient Heat Transfer Problem”, National Conference on Recent Research and Development in Core Disciplines of Engineering, Vadodara, Volume: 2, April 2015 [12] John Campbell, Castings (2nd Edition) [13] Peter Mullinger and Barrie Jenkins, Industrial and Process Furnaces (1st Edition) Elsevier Publication [14] A V K Suryanarayana, Fuels Furnaces Refractory and Pyrometry B S Publications [15] “Heat and Mass Transfer, A practical Approach” by Yunus A Cengel.
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