issue of heat losses from slug calorimeter cited in · fitting this data to slug loss model...

2Thermal & Fluids Analysis Workshop 2008 2

Outline

Arc jet descriptionSlug calorimeter descriptionIssue of heat losses from slug calorimeter cited in literatureIdealized slug calorimeter theory no losses and constant physical propertiesGeneral slug calorimeter theory with losses and variable heat capacity Slug Loss ModelSlug Loss Model applied to slug calorimeter data from one arc jet runFinite Element Analysis (FEA) model of slug calorimeter data from same arc jet runComparison of all modelsConclusions

3

Arc jet description

Thermal & Fluids Analysis Workshop 2008 3

Arc Jet Facility Simulation

Reentry FlightEnvironment

25x25 cm Test Body in the Arc Jet Plasma Stream

100 cm Diameter Nose Cap On a Reusable Vehicle

4

Slug calorimeter description

Thermal & Fluids Analysis Workshop 2008Figure Credit: ASTM Standard E457-08

4

Schematic of a Thermal Capacitance (Slug) Calorimeter

5


Thermal & Fluids Analysis Workshop 2008Figure Credit: ASTM Standard E457-08

5

Typical Temperature Time Curve for Slug Calorimeter

p b bp

Mc T Tq L c

A t t

6



Time Curve when heat & other losses are significant during heating phase

7

Slug calorimeter heat losses & correcting for them (quotes from literature)


The heat losses are usually hard to control in models with high-heat-flux conditions.

Diller, T. E., Advances in Heat Flux Measurements, Advances in Heat Transfer, Vol. 23, Academic Press, 1993, pp. 307-311

If more accurate results are required, the losses through the insulation layer should be modelled and accounted for by a correction term . . .

Childs, P. R. N., Greenwood, J. R., and Long, C. A., Heat flux measurement techniques, Proceedings of the Institution of Mechanical Engineers, Vol. 213, Part C, 1999, pp. 664-665.

8

Idealized slug calorimeter theory


Right circular cylinder made of copperInsulated at back face & around

circumferential areaSlug initially at uniform temperatureStarting at time = 0, constant heat flux

q is applied to front faceCoordinate x defined as zero at front

face and L at back faceProblem can be modeled as one

dimensional unsteady state heat transferAdditional simplifying assumption: all

physical properties are constant with temperature

9

Boundary value problem


2

2

pcT T

x k t

p

k

c

(0, )

( , )0

T t q

x kT L t

x

( ,0) oT x T

PDE for one dimensionalunsteady state heat transfer

Definition ofthermal diffusivity

Boundary conditions

Initial condition

10

Solution to PDE boundary value problem


2

( , )3 2ss o

p

qt qL qx qxv x t T

L c k Lk k

2

2 21

2 1( , ) cos

nt

L

n

qL n xw x t e

k n L

( , ) ( , ) ( , )ss

Overall solution steady state solution transient solution

T x t v x t w x t

22

2 21

2 1( , ) cos

3 2

nt

Lo

np

qt qL qx qx qL n xT x t T e

L c k Lk k k n L

( , )T x tconstant

t

11

Animation of solution for copper slugL= 1 cm, q= 2600 W/cm2, elapsed t= 0.3 s


Steady state solution

Response slowed down by a factor of 33

12



Transient solution


13



Overall solution


14

Response time equation


2

2

2ln

1R

indicated

input

Lt

q

q

Setting qindicated = 0 gives time for the heat to have just penetrated to the back side of the slug.

2

0.99 2

2ln

1 0.99R

Lt

For practical purposes, the response time calculated when qindicated/qinput = 0.99 should be sufficient elapsed time for the heat flux determination from the back face temperature to begin to be valid, and implies steady state.

15

Other useful equations


2

2 21

1 2 ( 1)( , )

6

nn tL

b on

q t qL qLT T L t T e

kL k k n

2

2 21

1 2 1(0, )

3

nt

Lf o

n

q t qL qLT T t T e

kL k k n

ave o

q tT T

kL

2

2

(2 1)

2 21

2 21

1 4 1

2 (2 1)

1 2 ( 1)

6

nt

Lf b

n

nn tL

ave bn

qL qLT T e

k k n

qL qLT T e

k k n

16

Summation terms approach zero with steady state


1( , )

6b o

q t qLT T L t T

kL k

1(0, )

3f o

q t qLT T t T

kL k

ave o

q tT T

kL

1

21

6

f b

ave b

qLT T

kqL

T Tk

17

Accounting for heat losses Slug Loss Model


A heat balance on the slug with losses gives

( . . )

ave o avepo

la

input output i e losses accumulation

T T dTqA Mc

R dt

6ave b

qLT T

k

6o b b

po la po la po la po

T T dTA Lq

Mc kR Mc R Mc R Mc dt

Getting equation in terms of Tb

18



Defining two constants

Differential equation can be written as

6

1

o

po la po la po

la po

TA La q

Mc kR Mc R Mc

bR Mc

bb

dTa bT

dt

1

1b t t

b b fit

a aT T e

b b

Which integrates to

19



Once data is fit to this

6

1

o

po la po la po

la po

TA La q

Mc kR Mc R Mc

bR Mc

1

1b t t

b b fit

a aT T e

b b

Rearrange this

16

po o

la

Mc a bTq

A L

kR A

To solve for q

20



Once you have this fit

By this equation

you also have

1

1( ) b t tb b fit

a aT t T e

b b

6ave b

qLT T

k

1

1( )6

b t tave b fit

a a qLT t T e

b b k

1

1

( ) ( ) b t tb aveb fit

dT t dT t ab T e

dt dt band

All analytical expressions

21



Now write energy balance equation with actual loss resistance and variable heat capacity

Variable heat capacity with T is obtained from the Shomateequation for copper

( ) ( )( ( ))ave o ave

p avel

T t T dT tqA Mc T t

R dt

2 32

2 12

4 73 4

6

( )

2.789933 10 4.421789 10

4.918152 10 2.19879 10

1.079706 10

p ave ave ave aveave

Ec T A BT CT DT

T

where

J JA x B x

kgK kgK

J JC x D x

kgK kgK

JKE x

kg

22



Solve energy balance equation for Rl, the actual loss resistance

Other useful equations

( )( )

( )( ( ))

ave ol

avep ave

T t TR t

dT tqA Mc T t

dt

( ( )) ( )( ) p b b

slopeTb

Mc T t dT tq t

A dt

( ( )) ( )( ) p ave ave

slopeTave

Mc T t dT tq t

A dt

( ( )) ( )( ) ( ) p ave ave

loss slopeTave

Mc T t dT tq t q q t q

A dt

( ( )) ( )( ) 1 p ave ave

Mc T t dT tFracLoss t

qA dt

23

Slug Loss Model (SLM) applied to one arc jet run (IHF187R025)


Back Face Temperature & Stagnation Pressure versus Time.

24

SLM applied to one arc jet run (IHF187R025)


38,925.7

kg

m385.2

Wk

mK385.615po

Jc

kgK

0.004529M kg

2 20.25 0.000047906A D m

0.010592M

L mA

2

0.99 2

2ln 0.538

1 0.99po

R

c Lt s

k

0.00781D m

25



Back Face Temperature & Stagnation Pressure versus Time.

t1

t2

26



Back Face Temperature versus Time data from t1 to t2.

1

1( ) b t tb b fit

a aT t T e

b bFit to this equation

27



Fitting this data to Slug Loss Model equation gives the best linear fit to Tb versus e-b(t-t1) when b = 0.29160 s-1, where the R2 value of the fit is maximized at 0.99999. The Solver function in an Excel spreadsheet was used to solve for b.

1intercept 766.76b t tb

Ka T vs e b

s

1

1 slope 660.32b t tb fit b

aT T vs e K

b

28



11.964la

po

KR

bMc W

2 226,005,000 2,600

16

po o

la

Mc a bT W Wq

A m cmL

kR A

This value is about 15% higher than the value of2,250 W/cm2 reported by the facility test engineers, where losses were not taken into account.

29



Back Face Temperature Fit Compared to Data.

30



Fit Compared to straight line.

31



Losses (per cm2 slug frontal area) versus Time.

32



Actual Loss Resistance versus Time.

33

Animation of a 4 Hemi Slug Calorimeter


34

FEA Model

A simple FEA model was created using COMSOL Multiphysics, COMSOL, Inc, Burlington, Massachusetts.The slug was modeled using 3D tetrahedral elements.Heat flux is applied to the top face using a smoothed Heaviside function (flc2hs) to create a ramp up and ramp down time.Losses occur through 0.6 mm diameter surface regions with a constant heat transfer coefficient h, to a constant holder temperature T0.The material copper is used using temperature dependent properties of heat capacity and thermal conductivity. 3 second simulation time with a 0.01 second time step.

Thermal & Fluids Analysis Workshop 2008

35

FEA Model


36

FEA Model

Various runs were performed by varying qinput, h and the duration of the pulse in order to match the data.A unique solution of qinput= 2,600 W/cm2 was found where the COMSOL solution closely agreed with the actual data. Sensitivity analysis showed this q value to be determinable to +/- 1%.A fringe plot of the temperature at t = 3 seconds was plotted to show the paths of the heat flow.Temperature was plotted versus time for the centers of the front and back faces of the slug.


37Thermal & Fluids Analysis Workshop 2008

Adjusting the results to actual data

38

FEA Model


39

COMSOL model with q = 2600 W/cm2 and actual data compared


40

COMSOL model with q = 2600 W/cm2 and actual data compared


41

Ideal PDE & COMSOL No Loss Const Phys Props


COMSOL VS Ideal PDE Comparison

0

200

400

600

800

1000

1200

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Ideal PDE SolutionCOMSOL Constant k & Cp% Difference

42

Ideal PDE & COMSOL No Loss Const Phys Props


43

COMSOL No Loss Const Phys Prop compared toLoss Const Phys Prop & No Loss Var. Phys Prop


44

Loss Const Phys Prop & No Loss Var. Phys Prop compared to COMSOL Loss Var. Phys Prop


45

COMSOL Loss Var. Phys Prop compared toSlug Loss Model


46

All Six cases


47

Conclusions

A mathematical model, The Slug Loss Model, was developed, which takes into account losses, where the temperature time slope takes the mathematical form of exponential decay.The Slug Loss Model was applied to slug calorimeter data from a high heat flux arc jet run.A FEA Model was also developed and run for various cases.Good agreement was shown between the Slug Loss Model and the FEA Model.


issue of heat losses from slug calorimeter cited in · fitting this data to slug loss model...

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