heat transfer through plane walls

A. INTRODUCTION

A heat transfer situation in which time is not a factor is designed as steady state. The consideration of heat transfer where time is not considered afford some simplification in the analysis. The governing equation for steady state conduction with internal generation is(Welty, 1978 and Holman, 1989)

V²T + = 0 ………………………………………………………(1)

Where :

VT : temperature gradient in vector form

q : heat flow vector

k : thermal conductivity vector

which is known as the Poisson equation, and for steady – state conduction without internal generation of heat, the Laplace equation applies :

V²T = 0 ……………………………………………………………...(2)

Both of the above equation apply to an isotropic medium, that is, one whose properties do not vary with direction, physical properties are also presumed independent of temperature.

The initial consideration is one – dimensional steady – state conduction without internal generation of energy. As just discussed. The Laplace equation applies to this case. A general form of Laplace equation in one dimension is:

= 0 …………………………………………………..(3)

Where

x : critical geometry in the direction of heat transfer, m

: temperature gradient a long x axis

: 0, 1, or 2 in rectangular, cylindrical, and spherical coordinate respectively

Plane Walls

In the case of he plane wall as shown in figure 1, equation (3) with i = 0 applies.

The equation and boundary conditions to be satisfied are

= 0…………………………………………………………………(4)

T(x) = T(0) = T₀ at x = 0

T(x) = T(L) = T_L at x = L

Where

T_o : temperature at x = 0, K

T_L : temperature at x = L, K

Equation (4) may be separated and integrated twice to yield

T(x) = c₁x + c₂ …………………………………………………………(5)

And the constants of integration c₁ and c₂ evaluated, by applying the boundary equation, to be

c_{1 =} and c_{2 =} T₀

When c₁ and c₂ are substituted into equation (5), the final expression for the temperature profile becomes

T(x) =

T(x) = _{………………………………………………………….(6)}

According to equation (6), the temperature variation in a plane wall under the condition specified is linear as shown in Figure 1.

The Fourier rate equation may be used to determined heat flux of heat flow rate in this case. The equation is repeated below, in scalar form in reference

q_x = …………………………………………………………...(7)

where

q_x : heat flux, W/m²

A : cross section area, m²

k : thermal conductivity, W/m K

Since, in the steady- state case, q, is constant, this equation may be separated and integrated directly as

giving

………………………………………………..(8)

Alternately, the temperature gradient dT/dx could been evaluated from equation (6) and substituted into equation (7) to achieve the identical result. These two alternate means of evaluating heat flux, either by direct integration of the Fourier rate equation or by- solving for temperature and substituting the temperature gradient expression into the rate equation, are both employed in subsequent examples. One approach may be simpler than another in certain case, but no general statement can be made in this regard.

The quantity kA/L, in equation (8), is the thermal conductance for a flat plate or wall. The reciprocal of this quantity, L/kA, is designated the thermal resistance.

B. OBJECTIVES

1. To assess the characteristic of the steady state heat transfer though the plane wall

2. To determine the rate of steady state heat transfer through the plane wall

C. MATERIALS AND EQUIPMENTS

Material are plane walls made out of aluminium. The setup of the experiment is illustrated in fig 2 comprises of Hybrid Recorder Yokogawa DR 130, a water bath at 100⁰C and thermocouples. Eleven observation point are inserted into the aluminium block and the distance between two observation points are 0.01 m

D. METHODS

1. Record the initial temperature at the observation point located as described in Fig 2

2. Pour boiling water at 100 ⁰C into the water both until the water surface reaches the height of the aluminium plane wall. Maintain the water at a constant temperature.

3. Record the temperature developed at the observation point in the interval of two minutes

E. RESULT AND DISCUSSION

Result from data experiment

Table 1

No	x/T	T0	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10
1	0	71,7	40,3	30,6	28,8	27,9	27,2	26,6	26,4	26,3	26,4	80,6
2	2	76,3	51,2	36,9	33,7	31,9	30,2	27,9	27	26,7	26,5	80,8
3	4	80	59,2	44,3	40,4	38,3	35,8	31,5	29,1	28,2	27,6	82,1
4	6	82,7	63,9	49,8	45,8	45,7	40,7	35,6	32,2	31	30	85,3
5	8	85,1	67,1	53,6	49,8	47,7	44,7	39,2	35,3	33,8	32,5	89,8
6	10	87,3	70,1	57	53,2	51,4	48,3	42,7	38,6	36,8	35,5	92,7
7	12	89,4	73,3	60,7	57	55,4	52,3	48,7	42,5	40,7	39,2	94,7
8	14	90,8	75,5	63,5	59,9	58,6	55,5	50	45,8	43,9	42,4	95,2
9	16	92,6	78,1	66,6	63,2	62	58,9	53,6	49,5	47,7	46	96,9
10	18	95	80,7	69,2	65,9	64,7	61,7	56,5	52,4	50,7	48,9	98,8
11	20	96,1	83,4	72,3	68,9	67,8	64,9	59,6	55,8	54	52,3	99,5
12	22	96,2	84,6	74,2	71	70,1	67,2	62,1	58,3	56,7	54,9	99,6
13	24	96,2	85,7	76,1	73,3	72,6	69,7	65	61,3	59,7	58	99,5
14	26	95,9	86	77,3	74,7	74,2	71,5	67	63,6	62,1	60,3	99,5
15	28	95,9	86,7	78,4	75,9	75,7	73,1	68,8	65,6	64,2	62,4	99,5
16	30	95,9	87,3	79,5	77,2	77,7	74,7	70,8	67,6	66,3	64,6	99,5
17	32	96,2	88,2	80,6	78,3	78,4	76	72,5	69,6	68,4	66,7	99,5
18	34	96,1	88,6	81,5	79,2	79,4	77	73,4	70,8	69,7	68,1	99,5
19	36	96	88,7	82	80	80,3	78,1	74,8	72,1	71	69,3	99,5
20	38	96,1	89	82,6	87	81,1	78,9	75,7	73,2	72,1	70,5	99,5
21	40	96,1	89,3	83,3	81,4	81,8	79,5	76,6	74,2	73,2	71,5	99,5
22	42	96,2	89,6	83,5	81,7	82,3	80,2	77,3	75	74,1	72,6	99,5
23	44	96	89,6	84	82,2	82,8	80,7	78	75,7	74,8	73,2	99,5
24	46	95,9	89,6	84,2	82,7	83,4	81,2	78,3	76,2	75,3	73,7	99,5
25	48	96,3	90,2	84,6	82,9	83,6	81,5	78,9	76,7	75,9	74,2	99,5
26	50	96,1	90,2	84,8	83,1	83,9	81,9	79,1	77,1	76,4	74,7	99,5
27	52	96,1	90,4	85,1	83,6	84,3	82,3	79,8	77,7	76,9	75,3	99,5
28	54	96,1	89,9	85,1	83,7	84,6	82,6	80,1	78,1	77,4	75,7	99,5
29	56	95,8	90,3	85,3	83,9	84,7	82,7	80,4	78,4	77,7	76,2	99,5
30	58	96,1	90,6	85,5	83,8	84,8	82,7	80,4	78,6	78	76,4	99,5
31	60	96,1	90,8	85,7	84,3	85,1	83,2	80,8	78,9	78,1	76,5	99,5

Table 2

No	q/A	T0	T1	T2	T3	T4	T5	T6	T7	T8	T9	T10
1	22250	80,6	79,71	78,82	77,93	77,04	76,15	75,26	74,37	73,48	72,59	71,7
2	11250	80,8	80,35	79,9	79,45	79	78,55	78,1	77,65	77,2	76,75	76,3
3	5250	82,1	81,89	81,68	81,47	81,26	81,05	80,84	80,63	80,42	80,21	80
4	6500	85,3	85,04	84,78	84,52	84,26	84	83,74	83,48	83,22	82,96	82,7
5	11750	89,8	89,33	88,86	88,39	87,92	87,45	86,98	86,51	86,04	85,57	85,1
6	13500	92,7	92,16	91,62	91,08	90,54	90	89,46	88,92	88,38	87,84	87,3
7	13250	94,7	94,17	93,64	93,11	92,58	92,05	91,52	90,99	90,46	89,93	89,4
8	11000	95,2	94,76	94,32	93,88	93,44	93	92,56	92,12	91,68	91,24	90,8
9	10750	96,9	96,47	96,04	95,61	95,18	94,75	94,32	93,89	93,46	93,03	92,6
10	9500	98,8	98,42	98,04	97,66	97,28	96,9	96,52	96,14	95,76	95,38	95
11	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
12	8500	99,6	99,26	98,92	98,58	98,24	97,9	97,56	97,22	96,88	96,54	96,2
13	8250	99,5	99,17	98,84	98,51	98,18	97,85	97,52	97,19	96,86	96,53	96,2
14	9000	99,5	99,14	98,78	98,42	98,06	97,7	97,34	96,98	96,62	96,26	95,9
15	9000	99,5	99,14	98,78	98,42	98,06	97,7	97,34	96,98	96,62	96,26	95,9
16	9000	99,5	99,14	98,78	98,42	98,06	97,7	97,34	96,98	96,62	96,26	95,9
17	8250	99,5	99,17	98,84	98,51	98,18	97,85	97,52	97,19	96,86	96,53	96,2
18	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
19	8750	99,5	99,15	98,8	98,45	98,1	97,75	97,4	97,05	96,7	96,35	96
20	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
21	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
22	8250	99,5	99,17	98,84	98,51	98,18	97,85	97,52	97,19	96,86	96,53	96,2
23	8750	99,5	99,15	98,8	98,45	98,1	97,75	97,4	97,05	96,7	96,35	96
24	9000	99,5	99,14	98,78	98,42	98,06	97,7	97,34	96,98	96,62	96,26	95,9
25	8000	99,5	99,18	98,86	98,54	98,22	97,9	97,58	97,26	96,94	96,62	96,3
26	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
27	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
28	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
29	9250	99,5	99,13	98,76	98,39	98,02	97,65	97,28	96,91	96,54	96,17	95,8
30	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1
31	8500	99,5	99,16	98,82	98,48	98,14	97,8	97,46	97,12	96,78	96,44	96,1

Contoh Perhitungan:

-

-

(untuk T1=0,01 m) (untuk T2=0,02 m)

(untuk interval menit ke-0)

- (untuk interval menit ke-2)

F. DISCUSSION

From this experiment we want to know about the heat transfer trough plane walls. If we see from first table, the temperature T0 until T10 increase linearly, because T10 is directly contact between the cable and water, so T10 is the highest among the others temperatures. On the other hand, in the second table we changed T10 and T0, so the highest temperature is T0. Then, T1 until T9 are gotten from the calculation that the formula is given from the laboratory manual.

As we seen in the graphic relation between thickness and temperature and graphic relation between flux and time, they all relatively getting down and down in several time. It means that the thickness of material can involve the temperature too if we heat the material. The outer surface will be colder than the inside. It’s all because of the temperature is absorb by every section. So, the heat will get down and down. The data from Hybrid Recorder Yokogawa DR 130 is different with the data from calculation.

G. CONCLUSION

From this experiment we can conclude that the students have understood about the heat transsfer through plane walls. It can be looked from the data and the result of the laboratory work.

H. REFERENCES

Holman, J.P. 1997. Heat Transfer. McGraw Hill Book Co., Singapore

Welty, J.R. 1978. Engineering Heat Transfer. John Wiley & Sons, New York, N.Y. USA

Laboratory Work Result Wednesday, March 24^th 2010

Thermodynamic and Heat Transfer Energy and Electrification of

Ahricultural Laboratory

CONDUCTION HEAT TRANSFER THROUGH PLANE WALLS

By :

Ahmad Eriska Dwi Hutama Putra

F14080122

DEPARTEMENT OF AGRICULTURAL ENGINEERING

FACULTY OF AGRICULTURAL TECHNOLOGY

BOGOR AGRICULTURE UNIVERSITY

2010