bessel

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08/09/2022 document.xls UNSTEADY STATE HEATING WITH HOT WATER 8/9/2022 Kern, p627 Input data Heat-up time h 4 Overall HTC Btu/h sq.ft °F U 38 Wt x Sp.ht Heated fluid heat capa Btu Mc 6206 = 6206 1 Heating fluid heat cap Btu/h WC 15000 = 15000 1 Starting batch tempera °F t1 70 Ending batch temperatu °F t2 160 Heating fluid inlet te °F T1 185 Heat transfer area A sq.ft 67.81229 K 1.18743 LHS ln(T1-t1/T1-t2) 1.526056 solve for these two RHS WC*(K-1)*t/Mc/K 1.526055 rows being equal

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Page 1: Bessel

04/17/2023 document.xls

UNSTEADY STATE HEATING WITH HOT WATER

4/17/2023 Kern, p627

Input data

Heat-up time h 4

Overall HTC Btu/h sq.ft °F U 38 Wt x Sp.ht

Heated fluid heat capacity Btu Mc 6206 = 6206 1

Heating fluid heat capacity Btu/h WC 15000 = 15000 1

Starting batch temperature °F t1 70

Ending batch temperature °F t2 160

Heating fluid inlet temp °F T1 185

Heat transfer area A sq.ft 67.81229

K 1.18743

LHS ln(T1-t1/T1-t2) 1.526056 solve for these two

RHS WC*(K-1)*t/Mc/K 1.526055 rows being equal

Page 2: Bessel

04/17/2023 document.xls

Time to heat a tank with steam at constant temperature

Tank volume gal 7200 Heat-up time hr 4.75

Density pcf 62.4 Max heat transfer rate Btu/h 900000

Specific heat Btu/lbF 1

Coil HTC Btu/sf.lb.F 45

Coil area sf 100

Start temp F 80

End temp F 140

Steam temp F 280

Page 3: Bessel

Page

Transient heating coefficients

from AIChE 1931 p311 - 'The drying of porous solids - diffusion calculations' by A.B.Newman

Enter data in yellow cells, and run solver.

Bessel roots 1 2

root 2.4048 5.5201

squared 5.783063 30.4715

Container dia 2r in 3

Container ht 2l in 4.22

Specific ht Cp Btu/lbF 1

Thermal cond k Btu/hsfF/f 0.25

Density s lb/cuf 62.4

Time t min 4

Diffusivity D k/Cps 0.004006

Radial kt/r^2 1.066667

Longit kt/l^2 0.539071

Coefficients

A' 0.003964

R' 0.001448

Temp start To F 270

Temp surface Ti F 185

End average Tav F 185.0005 Assuming negligible surface resistance

Including surface resistance

Outside HTC h Btu/hsfF 50

Radial mr k/rh 0.04 Int sol #NAME? #NAME?

Longit ml k/lh 0.028436 Int sol #NAME? #NAME?

Char # rad w 2.310798 5.306801 first and second roots

Char # long u 2.337504 5.366504 needed if x<0.2

X radial 0.017094

X longit 0.008639

a radial #NAME? #NAME? first term only

a longit #NAME? #NAME?

Y #NAME?

Temp centre F #NAME?

A B C D E F G1

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Page 4: Bessel

Page

from AIChE 1931 p311 - 'The drying of porous solids - diffusion calculations' by A.B.Newman

3 4 pi^2

8.654 11.792 9.869604

74.89172 139.0513

Assuming negligible surface resistance

first term only

H I J1

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