Unsteady-state heat transfer

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Heat Transfer

Nur Istianah-THP-FTP-UB-2016

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Unsteady-state heat transfer

Temperature is changing with time, it is a function of both location and time

It was in such as process: food pasteurization, sterilization, refrigeration/chilling/cooling

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Introduction

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• In another words, in unsteady transient heat conduction, temperature is a function of both time and spatial coordinates

 By using Fourier’s Equation governs the temperature response of a body, and Thermal Energy equation (first law of thermodynamics for closed system), one can fined the thermal conductivity of any body by recording temperature change of its center with the time.

 Rate of heat transfer by conduction from the center of the body out side or in opposite direction can be describe by Fourier’s Law

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Unsteady state heat transfer

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• When the body is a metal semi infinite slab or cylinder or sphere, for one-dimensional case the combination equation is:

Where for slab m=0 for cylinder m=1

for sphere m=2

From thermodynamics definition of the thermal diffusivity

The general equation may be modified to:

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Unsteady state heat transfer

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Where



 T_i

T_s

T_ Bi<0.1

Bi=1 Bi>>1

This Figure shows Meaning of Biot number value (the relation between Biot number value and the temperature gradient of any solid object).

Convection

T_s T_

Conduction

T_i T_s

R

T_i T_s T_

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Biot Number

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Three cases for unsteady-state heat transfer

• N

_Bi

< 0.1 : negligible internal resistance to heat transfer

• 0.1 < N

_Bi

< 40 : finite internal and surface resistance to heat transfer

• N

_Bi

> 40 : negligible surface resistance to heat transfer

Nur Istianah-THP-FTP-UB-2016 14

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• This indicates that the thermal resistance is negligible compared to the convection resistance and T_i  T_s and the solid may have a uniform temperature.

• This phenomenon is called (Lumped Thermal Capacity).

• Where in this case the heat transfer by convection from the surface will balance the change of the internal energy of the solid.

At initial conditions t = 0 T = T_i

the integration of of the above eq.

T -T_¥

T_i -T_¥ = e^-

h rC_pRt

1 G

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Low Biot Number less than 0.1

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This equation shows that the solid body temperature approaches the surrounding temperature T_ exponentially, which means that the solid temperature change rapidly at the beginning and slowdown after.

For a big value of (1/) the solid reach T_ in very short time (very high

conducting material).

𝑇 − 𝑇_∞

𝑇_𝑖 − 𝑇_∞ = 𝑒⁻^Γ^𝑡

G =r^CpR

h Time constant

Fourier modules (Fo)

• If we rearrange the exponent term in the above equation

𝑡

Γ = ℎ𝑡

𝜌𝐶_𝑝𝑅 = ℎ𝑅

𝑘 𝑘

𝜌𝐶_𝑝𝑅² 𝑡

Fourier number (Fo)

Fo =

a

R² t

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Contoh soal [4]

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Y = ϴ

X = Bi*Ʈ

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0.1 <N_Bi < 40

Finite internal and surface resistances to heat transfer

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Cp k R X t

T T

T Y T

 







 

2 0 1

1 T₁ = suhu medium (K)

T₀ = suhu awal (K) T = suhu akhir (K)

 = difusivitas termal t = waktu (detik)

k = konduktivitas panas (W/m.K)

h = koefisien transfer panas (W/m².K)

 = densitas (kg/m³)

c_p = panas spesifik (J/kg.K) x = posisi dari pusat (m)

R = dimensi relatif (1/2 ketebalan) (m)

) , ,

(X n m f

Y 

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Unsteady State (dimensionless)

R n x

hR m k



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(T_a–T) (T_a–T_i)

N_Fo = t / D² 1/N_Bi

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N_Fo = t / D² 1/N_Bi

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N_Fo = t / D² 1/N_Bi

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• For calculation of temperature at any position of the object, Gurney-Lurie Chart can be used.

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Gurney-Lurie Chart

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2

x1

X t



position t

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Contoh soal [5]

Daging berbentuk persegi dengan ketebalan 2.54 cm mempunyai suhu awal 10°C dimasak dalam oven dengan suhu 177°C sehingga suhu pusat mencapai 121°C. Koefisien konveksi diasumsikan konstan sebesar 25.6 W/m².K. Hitung waktu yang diperlukan!

Konduktivitas panas sebesar 0.69 W/m.K dan difusivitas panas sebesar 5.85x10^-4 m²/jam.

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Cp k R X t

T T

T Y T

 







 

2 0 1

1 T₁ = suhu medium (K)

T₀ = suhu awal (K) T = suhu akhir (K)

 = difusivitas termal t = waktu (detik)

k = konduktivitas panas (W/m.K)

h = koefisien transfer panas (W/m².K)

 = densitas (kg/m³)

c_p = panas spesifik (J/kg.K) x = posisi dari pusat (m)

R = dimensi relatif (1/2 ketebalan) (m)

) , ,

(X n m f

Y 

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Unsteady State (dimensionless)

R n x

hR m k



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Contoh soal [5]

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Produk butter dengan ketebalan 92,4 mm yang mempunyai suhu 277,6 K ditempatkan dalam ruangan bersuhu 297,1 K. Bagian samping dan bawah kontainer butter dianggap terisolasi, sedangkan bagian permukaan atas berhubungan dengan udara luar. Koefisien konveksi sebesar 8,52 W/m². Hitung suhu pada permukaan atas, pada jarak 25,4 mm di bawah permukaan dan pada bagian tengah dari butter setelah 5 jam dibiarkan di udara luar.

k butter 0,197 W/m.K, cp = 2300 J/kg.K,  = 998 kg/m³

Latihan soal [2]

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9.24 cm

x= 4.62 cm

x= 4.62-

2.54=2.08cm

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Finite objects

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Finite object

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Y _x = T₁ - T_x dengan X_x dan m_x T₁ – T₀

Y _y = T₁ - T_y dengan X_y dan m_y T₁ – T₀

Y _z = T₁ - T_z dengan X_z dan m_z T₁ – T₀

Y _x,y,z = (Y_x)(Y_y)(Y_{z )})= T₁ – T_x,y,z T₁ – T₀

Finite object (Two or three dimension)

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Kaleng silinder berisi softdrink mempunyai diameter 4,8 cm dan suhu awal 27C. Kaleng ini disterilkan dalam retort dan dikenai uap dengan suhu 177C. Hitung suhu di pusat kaleng selama pemanasan 45 menit jika bagian atas dan bawah kaleng terisolasi.

Koefisien transfer panas dari uap sebesar 4540 W/m².K. Sifat fisik dari softdrink adalah sebagai berikut; k 0,48 W/m.K dan  2,007 x 10^-7 m²/s.

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Contoh soal [6]

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Contoh soal [6]

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Kaleng silinder berisi puree mempunyai diameter 2,68 in dan tinggi 4 in serta suhu 85 F. Kaleng ini disterilkan dalam retort dan dikenai uap dengan suhu 240 F.

Hitung suhu di pusat kaleng selama pemanasan 45 menit.

Koefisien transfer panas dari uap sebesar 4542 W/m².K. Sifat fisik dari puree adalah sebagai berikut;

k 0,830 W/m.K dan  2,007 x 10^-7 m²/s.

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• Estimate the time when temperature at the geometric center of a 6 cm diameter apple held in 2C water stream reaches 3C. The initial uniform temperature of the apple is 15C. The convective heat transfer coefficient in water surrounding the apple is 50 W/m²C. The properties of the apple are thermal conductivity k = 0.355 W/mC, specific heat Cp = 3.6 kJ/kgC, and density = 820 kg/m³.

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• Estimate the temperature at the geometric center of a food product contained in a 303X406 can exposed to boiling water at 100C for 30 min. The product is assumed to heat and cool by conduction. The initial uniform temperature of product is 35C. The properties of the food are thermal conductivity k = 0.34 W/mC, specific heat Cp = 3.5 kJ/kgC, and density  = 900 kg/m³. The convective heat transfer coefficient for boiling water is estimated to be 2000 W/m²C.

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1. Karkas daging mempunyai nilai  1073 kg/m³, cp 3,48 kJ/kg.K, dan k 0,498 W/m.K. Daging berbentuk lempeng mempunyai ketebalan 0,203 m dan suhu awal 37.8 C didinginkan sehingga suhu pusat mencapai 10 C.

Udara dingin bersuhu 1,7 C dan mempunyai nilai h 39,7 W/m².K digunakan untuk mendinginkan.

Hitung waktu yang diperlukan.

TUGAS

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2. Daging berbentuk persegi panjang dengan ketebalan 4 cm dengan suhu awal 8C dimasak dengan menggunakan oven bersuhu 185C hingga mencapai suhu pusat 121 C.

Koefisien konveksi diasumsikan konstan dengan nilai sebesar 25,6 W/m².K. Nilai konduktivitas panas sebesar 0,69 W/m.K dan nilai difusivitas panas () sebesar 5,85 x 10^-4 m²/jam.

• Hitung waktu yang diperlukan untuk proses di atas!

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3. Pipa baja berisi uap air mempunyai diameter luar 95 mm dan nilai k sebesar 45 W/m.K. Pipa ini dilapisi dengan 75 mm isolator yang mempunyai nilai k 0,043 W/m.K. Dua termokopel ditempatkan pada antarmuka antara dinding pipa dan isolator serta pada permukaan luar isolator dan menunjukkan suhu berturut-turut 125C dan 35C.

• Hitung kehilangan panas yang terjadi per meter pipa.

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