Question

Thermal energy storage systems commonly involve a packed bed of solid spheres through which a hot gas flows when the system is being charged. In the charging process heat is transferred from the hot gas to the spheres and it increases the thermal energy stored in the spheres. Here are some variables that are important - the convection coefficient between the particle surface and the gas is h (W/m2 K) and the particle thermophysical properties are k (W/m K), Cp (J/kg K) and p (kg/m3). The radius of the particle is R (m). Initially the particles are at a temperature Ti (°C). The gas flow is started at t=0, and the gas temperature is at T. (°C) - can be assumed a constant with time. Let the gas flow through the particles for a time t (s). Develop an expression for the total heat transferred to a particle Q (units of J) in this time t (In terms of the variables given). Assume lumped system analysis holds. What is the maximum possible thermal energy (Qmax) a particle can accumulate?

Answer 1

Given :- ☺ Convection coefficient between particle Surface and gas = h (W/m²-k). w Particle thermal conductivity = k. (W/m-K) ③ Particle specific heat capacity = cp (J/kg -K) 6 Particle density = 8 (kg/m3), od 0 Radius of particle = R (m). @ Initial temperature of particle = Ti (°C). € Gas temperature = Too (°C) & Consider lumped systems analysis. or To find : a Expression particle in terms for total heat transferred to Q (J) in this time t of given variables. Max possible particle. thermal energy (Qmax) for a - Solution : for a small time interval, the particle temperature vises by a small amount. Let us say for a time at the temperature rise of the particle is 'dT'. By energy balance, the heat transferred into the particle = energy stored in the particle by rising its temperature.

Given :- ☺ Convection coefficient between particle Surface and gas = h (W/m²-k). w Particle thermal conductivity = k. (W/m-K) ③ Particle specific heat capacity = cp (J/kg -K) 6 Particle density = 8 (kg/m3), od 0 Radius of particle = R (m). @ Initial temperature of particle = Ti (°C). € Gas temperature = Too (°C) & Consider lumped systems analysis. or To find : a Expression particle in terms for total heat transferred to Q (J) in this time t of given variables. Max possible particle. thermal energy (Qmax) for a - Solution : for a small time interval, the particle temperature vises by a small amount. Let us say for a time at the temperature rise of the particle is 'dT'. By energy balance, the heat transferred into the particle = energy stored in the particle by rising its temperature.
​​​​​
Given :- ☺ Convection coefficient between particle Surface and gas = h (W/m²-k). w Particle thermal conductivity = k. (W/m-K) ③ Particle specific heat capacity = cp (J/kg -K) 6 Particle density = 8 (kg/m3), od 0 Radius of particle = R (m). @ Initial temperature of particle = Ti (°C). € Gas temperature = Too (°C) & Consider lumped systems analysis. or To find : a Expression particle in terms for total heat transferred to Q (J) in this time t of given variables. Max possible particle. thermal energy (Qmax) for a - Solution : for a small time interval, the particle temperature vises by a small amount. Let us say for a time at the temperature rise of the particle is 'dT'. By energy balance, the heat transferred into the particle = energy stored in the particle by rising its temperature.

• Therefore, heat transferred by convection through gas = increase in energy of the particle. hx (Area of particle) x ( Top -T) x dt = (mass of particle) & Cp x dT. - Area of particle = 415 x R². mass of particle = 8 x Volume of particle (v). m = p XV. Since dT can be expressed as dt = dl T- Too), where T is the temperature of pastice at any instant of time t. .. dT =-hx (45R² x dt... T-To fV Cp - - - Integrating the above equation from to any time t = t * at which t=0; T = Ti. T = TLt) gives - lu ITLE) - Too 7 Ti - To J io hx(40R² x at x3 8x4 TL R² x Cp. - lu Tlt) - Tao 1 = - 3ht [ Ti - Too J 4TRC Taking exponential on both sides, ITLE) To = é of where, b= 3h Ti- To - é s where 4TRC

• Once the temperature Tlt) for a given time. is available from egn , the total heat transferred to a particle, & is given by Newton's law of cooling as follows: - Q = kx (Area of particle) × [ TLE) - To] Q = 4TR ² h [TLE) - Too] • in the energy This is also equal to the increase of the particle Q = m Cp [TLE) - Ti] Q = 4 FIR Cp [TLt) - Ti] • The man possible heat transfer will-occur when : there is maximum temperature difference between the particle and the gas which is at time , t=0 wall i & max = IR max = mc [ Tp - Ti ] 4 TL P R col To - Ti]