Temperature of Body by Lumped Heat Capacity Method Calculator

✖The Heat Transfer Coefficient is the heat transferred per unit area per kelvin. Thus area is included in the equation as it represents the area over which the transfer of heat takes place.ⓘ Heat Transfer Coefficient [h]			+10% -10%
✖Surface Area for Convection is defined as the surface area of object which is in the process of heat transfer.ⓘ Surface Area for Convection [A_c]			+10% -10%
✖Time Constant is defined as the total time taken for a body to attain final temperature from initial temperature.ⓘ Time Constant [𝜏]			+10% -10%
✖Density of Body is the physical quantity that expresses the relationship between its mass and its volume.ⓘ Density of Body [ρ_B]			+10% -10%
✖Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount.ⓘ Specific Heat Capacity [c]			+10% -10%
✖Volume of Object is the amount of space that a substance or object occupies or that is enclosed within a container.ⓘ Volume of Object [V]			+10% -10%
✖The Initial temperature of Object is defined as the measure of heat under initial state or conditions.ⓘ Initial Temperature of Object [T₀]			+10% -10%
✖Temperature of Bulk Fluid is defined as the temperature of bulk fluid or fluid at given instant measured using thermometer.ⓘ Temperature of Bulk Fluid [T_∞]			+10% -10%

✖Temperature at Any Time T is defined as the temperature of an object at any given time t measured using thermometer.ⓘ Temperature of Body by Lumped Heat Capacity Method [T]

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Formula

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Temperature of Body by Lumped Heat Capacity Method

Formula

`"T" = (exp((-"h"*"A"_{"c"}*"𝜏")/("ρ"_{"B"}*"c"*"V")))*("T"_{"0"}-"T"_{"∞"})+"T"_{"∞"}`

Example

`"556.0486K"=(exp((-"10W/m²*K"*"0.00785m²"*"1937s")/("15kg/m³"*"1.5J/(kg*K)"*"6.541m³")))*("887.36K"-"373K")+"373K"`

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Temperature of Body by Lumped Heat Capacity Method Solution

STEP 0: Pre-Calculation Summary

Formula Used

Temperature at Any Time T = (exp((-Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object)))*(Initial Temperature of Object-Temperature of Bulk Fluid)+Temperature of Bulk Fluid
T = (exp((-h*A_c*𝜏)/(ρ_B*c*V)))*(T₀-T_∞)+T_∞
This formula uses 1 Functions, 9 Variables

Functions Used

exp - n an exponential function, the value of the function changes by a constant factor for every unit change in the independent variable., exp(Number)

Variables Used

Temperature at Any Time T - (Measured in Kelvin) - Temperature at Any Time T is defined as the temperature of an object at any given time t measured using thermometer.
Heat Transfer Coefficient - (Measured in Watt per Square Meter per Kelvin) - The Heat Transfer Coefficient is the heat transferred per unit area per kelvin. Thus area is included in the equation as it represents the area over which the transfer of heat takes place.
Surface Area for Convection - (Measured in Square Meter) - Surface Area for Convection is defined as the surface area of object which is in the process of heat transfer.
Time Constant - (Measured in Second) - Time Constant is defined as the total time taken for a body to attain final temperature from initial temperature.
Density of Body - (Measured in Kilogram per Cubic Meter) - Density of Body is the physical quantity that expresses the relationship between its mass and its volume.
Specific Heat Capacity - (Measured in Joule per Kilogram per K) - Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount.
Volume of Object - (Measured in Cubic Meter) - Volume of Object is the amount of space that a substance or object occupies or that is enclosed within a container.
Initial Temperature of Object - (Measured in Kelvin) - The Initial temperature of Object is defined as the measure of heat under initial state or conditions.
Temperature of Bulk Fluid - (Measured in Kelvin) - Temperature of Bulk Fluid is defined as the temperature of bulk fluid or fluid at given instant measured using thermometer.

STEP 1: Convert Input(s) to Base Unit

Heat Transfer Coefficient: 10 Watt per Square Meter per Kelvin --> 10 Watt per Square Meter per Kelvin No Conversion Required
Surface Area for Convection: 0.00785 Square Meter --> 0.00785 Square Meter No Conversion Required
Time Constant: 1937 Second --> 1937 Second No Conversion Required
Density of Body: 15 Kilogram per Cubic Meter --> 15 Kilogram per Cubic Meter No Conversion Required
Specific Heat Capacity: 1.5 Joule per Kilogram per K --> 1.5 Joule per Kilogram per K No Conversion Required
Volume of Object: 6.541 Cubic Meter --> 6.541 Cubic Meter No Conversion Required
Initial Temperature of Object: 887.36 Kelvin --> 887.36 Kelvin No Conversion Required
Temperature of Bulk Fluid: 373 Kelvin --> 373 Kelvin No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

T = (exp((-h*A_c*𝜏)/(ρ_B*c*V)))*(T₀-T_∞)+T_∞ --> (exp((-10*0.00785*1937)/(15*1.5*6.541)))*(887.36-373)+373

Evaluating ... ...

T = 556.048556063287

STEP 3: Convert Result to Output's Unit

556.048556063287 Kelvin --> No Conversion Required

FINAL ANSWER

556.048556063287 ≈ 556.0486 Kelvin <-- Temperature at Any Time T

(Calculation completed in 00.020 seconds)

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University School of Chemical Technology-USCT (GGSIPU), New Delhi

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< 18 Unsteady State Heat Conduction Calculators

Temperature Response of Instantaneous Energy Pulse in Semi Infinite Solid

Go Temperature at Any Time T = Initial Temperature of Solid+(Heat Energy/(Area*Density of Body*Specific Heat Capacity*(pi*Thermal Diffusivity*Time Constant)^(0.5)))*exp((-Depth of Semi Infinite Solid^2)/(4*Thermal Diffusivity*Time Constant))

Time Taken by Object for Heating or Cooling by Lumped Heat Capacity Method

Go Time Constant = ((-Density of Body*Specific Heat Capacity*Volume of Object)/(Heat Transfer Coefficient*Surface Area for Convection))*ln((Temperature at Any Time T-Temperature of Bulk Fluid)/(Initial Temperature of Object-Temperature of Bulk Fluid))

Initial Temperature of Body by Lumped Heat Capacity Method

Go Initial Temperature of Object = (Temperature at Any Time T-Temperature of Bulk Fluid)/(exp((-Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object)))+Temperature of Bulk Fluid

Temperature of Body by Lumped Heat Capacity Method

Go Temperature at Any Time T = (exp((-Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object)))*(Initial Temperature of Object-Temperature of Bulk Fluid)+Temperature of Bulk Fluid

Temperature Response of Instantaneous Energy Pulse in Semi Infinite Solid at Surface

Go Temperature at Any Time T = Initial Temperature of Solid+(Heat Energy/(Area*Density of Body*Specific Heat Capacity*(pi*Thermal Diffusivity*Time Constant)^(0.5)))

Fourier Number given Heat Transfer Coefficient and Time Constant

Go Fourier Number = (Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object*Biot Number)

Biot Number given Heat Transfer Coefficient and Time Constant

Go Biot Number = (Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object*Fourier Number)

Fourier Number using Biot Number

Go Fourier Number = (-1/(Biot Number))*ln((Temperature at Any Time T-Temperature of Bulk Fluid)/(Initial Temperature of Object-Temperature of Bulk Fluid))

Biot Number using Fourier Number

Go Biot Number = (-1/Fourier Number)*ln((Temperature at Any Time T-Temperature of Bulk Fluid)/(Initial Temperature of Object-Temperature of Bulk Fluid))

Biot Number given Characteristic Dimension and Fourier Number

Go Biot Number = (Heat Transfer Coefficient*Time Constant)/(Density of Body*Specific Heat Capacity*Characteristic Dimension*Fourier Number)

Fourier Number given Characteristic Dimension and Biot Number

Go Fourier Number = (Heat Transfer Coefficient*Time Constant)/(Density of Body*Specific Heat Capacity*Characteristic Dimension*Biot Number)

Initial Internal Energy Content of Body in Reference to Environment Temperature

Go Initial Energy Content = Density of Body*Specific Heat Capacity*Volume of Object*(Initial Temperature of Solid-Ambient Temperature)

Fourier Number using Thermal Conductivity

Go Fourier Number = ((Thermal Conductivity*Characteristic Time)/(Density of Body*Specific Heat Capacity*(Characteristic Dimension^2)))

Time Constant of Thermal System

Go Time Constant = (Density of Body*Specific Heat Capacity*Volume of Object)/(Heat Transfer Coefficient*Surface Area for Convection)

Capacitance of Thermal System by Lumped Heat Capacity Method

Go Capacitance of Thermal System = Density of Body*Specific Heat Capacity*Volume of Object

Fourier Number

Go Fourier Number = (Thermal Diffusivity*Characteristic Time)/(Characteristic Dimension^2)

Biot Number using Heat Transfer Coefficient

Go Biot Number = (Heat Transfer Coefficient*Thickness of Wall)/Thermal Conductivity

Thermal Conductivity given Biot Number

Go Thermal Conductivity = (Heat Transfer Coefficient*Thickness of Wall)/Biot Number

Temperature of Body by Lumped Heat Capacity Method Formula

What is Unsteady State Heat Transfer?

Unsteady State Heat Transfer refers to the heat transfer process in which a system's temperature changes with time. This type of heat transfer can happen in different forms, such as conduction, convection, and radiation. It occurs in various systems, including solid materials, fluids, and gases. The heat transfer rate in an unsteady state is directly proportional to the rate of temperature change. This means that the heat transfer rate is not constant and can vary over time. It's an important aspect in the design and optimization of thermal systems, and understanding this process is crucial in many research areas, such as combustion, electronics, and aerospace.

What is Lumped Parameter Model?

Interior temperatures of some bodies remain essentially uniform at all times during a heat
transfer process. The temperature of such bodies are only a function of time, T = T(t). The
heat transfer analysis based on this idealization is called lumped system analysis.

How to Calculate Temperature of Body by Lumped Heat Capacity Method?

Temperature of Body by Lumped Heat Capacity Method calculator uses Temperature at Any Time T = (exp((-Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object)))*(Initial Temperature of Object-Temperature of Bulk Fluid)+Temperature of Bulk Fluid to calculate the Temperature at Any Time T, The Temperature of Body by Lumped Heat Capacity Method formula is defined as the function of heat transfer coefficient, surface area of convection, density of object, specific heat capacity of object, volume of body, initial temperature, temperature of convection environment, time taken for temperature change. This type of analysis is called the lumped-heat-capacity method. Such systems are obviously idealized because a temperature gradient must exist in a material if heat is to be conducted into or out of the material. In general, the smaller the physical size of the body, the more realistic the assumption of a uniform temperature throughout; in the limit a differential volume could be employed as in the derivation of the general heat-conduction equation. Temperature at Any Time T is denoted by T symbol.

How to calculate Temperature of Body by Lumped Heat Capacity Method using this online calculator? To use this online calculator for Temperature of Body by Lumped Heat Capacity Method, enter Heat Transfer Coefficient (h), Surface Area for Convection (A_c), Time Constant (𝜏), Density of Body (ρ_B), Specific Heat Capacity (c), Volume of Object (V), Initial Temperature of Object (T₀) & Temperature of Bulk Fluid (T_∞) and hit the calculate button. Here is how the Temperature of Body by Lumped Heat Capacity Method calculation can be explained with given input values -> 556.0486 = (exp((-10*0.00785*1937)/(15*1.5*6.541)))*(887.36-373)+373.

FAQ

What is Temperature of Body by Lumped Heat Capacity Method?

The Temperature of Body by Lumped Heat Capacity Method formula is defined as the function of heat transfer coefficient, surface area of convection, density of object, specific heat capacity of object, volume of body, initial temperature, temperature of convection environment, time taken for temperature change. This type of analysis is called the lumped-heat-capacity method. Such systems are obviously idealized because a temperature gradient must exist in a material if heat is to be conducted into or out of the material. In general, the smaller the physical size of the body, the more realistic the assumption of a uniform temperature throughout; in the limit a differential volume could be employed as in the derivation of the general heat-conduction equation and is represented as T = (exp((-h*A_c*𝜏)/(ρ_B*c*V)))*(T₀-T_∞)+T_∞ or Temperature at Any Time T = (exp((-Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object)))*(Initial Temperature of Object-Temperature of Bulk Fluid)+Temperature of Bulk Fluid. The Heat Transfer Coefficient is the heat transferred per unit area per kelvin. Thus area is included in the equation as it represents the area over which the transfer of heat takes place, Surface Area for Convection is defined as the surface area of object which is in the process of heat transfer, Time Constant is defined as the total time taken for a body to attain final temperature from initial temperature, Density of Body is the physical quantity that expresses the relationship between its mass and its volume, Specific Heat Capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount, Volume of Object is the amount of space that a substance or object occupies or that is enclosed within a container, The Initial temperature of Object is defined as the measure of heat under initial state or conditions & Temperature of Bulk Fluid is defined as the temperature of bulk fluid or fluid at given instant measured using thermometer.

How to calculate Temperature of Body by Lumped Heat Capacity Method?

The Temperature of Body by Lumped Heat Capacity Method formula is defined as the function of heat transfer coefficient, surface area of convection, density of object, specific heat capacity of object, volume of body, initial temperature, temperature of convection environment, time taken for temperature change. This type of analysis is called the lumped-heat-capacity method. Such systems are obviously idealized because a temperature gradient must exist in a material if heat is to be conducted into or out of the material. In general, the smaller the physical size of the body, the more realistic the assumption of a uniform temperature throughout; in the limit a differential volume could be employed as in the derivation of the general heat-conduction equation is calculated using Temperature at Any Time T = (exp((-Heat Transfer Coefficient*Surface Area for Convection*Time Constant)/(Density of Body*Specific Heat Capacity*Volume of Object)))*(Initial Temperature of Object-Temperature of Bulk Fluid)+Temperature of Bulk Fluid. To calculate Temperature of Body by Lumped Heat Capacity Method, you need Heat Transfer Coefficient (h), Surface Area for Convection (A_c), Time Constant (𝜏), Density of Body (ρ_B), Specific Heat Capacity (c), Volume of Object (V), Initial Temperature of Object (T₀) & Temperature of Bulk Fluid (T_∞). With our tool, you need to enter the respective value for Heat Transfer Coefficient, Surface Area for Convection, Time Constant, Density of Body, Specific Heat Capacity, Volume of Object, Initial Temperature of Object & Temperature of Bulk Fluid and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

How many ways are there to calculate Temperature at Any Time T?

In this formula, Temperature at Any Time T uses Heat Transfer Coefficient, Surface Area for Convection, Time Constant, Density of Body, Specific Heat Capacity, Volume of Object, Initial Temperature of Object & Temperature of Bulk Fluid. We can use 2 other way(s) to calculate the same, which is/are as follows -

Temperature at Any Time T = Initial Temperature of Solid+(Heat Energy/(Area*Density of Body*Specific Heat Capacity*(pi*Thermal Diffusivity*Time Constant)^(0.5)))*exp((-Depth of Semi Infinite Solid^2)/(4*Thermal Diffusivity*Time Constant))
Temperature at Any Time T = Initial Temperature of Solid+(Heat Energy/(Area*Density of Body*Specific Heat Capacity*(pi*Thermal Diffusivity*Time Constant)^(0.5)))

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